AE4M33RZN, Fuzzy logic: Fuzzy relations
74
AE4M33RZN, Fuzzy logic: Fuzzy relations Radomír Černoch [email protected] 9/11/2015 Faculty of Electrical Engineering, CTU in Prague 201 / 241 Relations
Transcript of AE4M33RZN, Fuzzy logic: Fuzzy relations
AE4M33RZN Fuzzy logicFuzzy relations
Radomiacuter Černochradomircernochfelcvutcz
9112015
Faculty of Electrical Engineering CTU in Prague
201 241 Relations
Plan of the lecture
Properties of fuzzy setsFuzzy implication and fuzzy propertiesFuzzy set inclusion and crisp predicates
Intermission Probabilistic vs fuzzy
Binary fuzzy relationsQuick revision of crisp relationsFuzzyfication of crisp relationsProjection and cylindrical extensionComposition of fuzzy relationsProperties of fuzzy relationsProperties of fuzzy composition
Extensions
Biblopgraphy202 241 Relations
Organizational
bull Last week (2 weeks from now) there will be a short test (max 5points) during the tutorials
bull This week we are having the last theoretical lecture
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Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
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Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
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Residue implication
Despite the lack of a uniform definition of fuzzy implication there is auseful class of implications
Defintion
The R-implication (residuum bdquoreziduovanaacute implikaceldquo) is a functionobtained from a fuzzy T-norm
120572 1113699rArr∘ 120573 = sup120574 | 120572 and∘ 120574 le 120573 (RI)
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R-implication Examples (1)
Standard implication (Goumldel) isderived from (RI) using thestandard cojunctionand1113700
120572 1113699rArr1113700120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573 otherwise
(3)
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R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
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R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
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R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
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R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
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R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
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S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
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S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
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Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
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Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
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Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
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Generalized fuzzy inclusion Example
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Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
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Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
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Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
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Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
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Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
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Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
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Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
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Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
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Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
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Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
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Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
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Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
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Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
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Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
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Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
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Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
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Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Plan of the lecture
Properties of fuzzy setsFuzzy implication and fuzzy propertiesFuzzy set inclusion and crisp predicates
Intermission Probabilistic vs fuzzy
Binary fuzzy relationsQuick revision of crisp relationsFuzzyfication of crisp relationsProjection and cylindrical extensionComposition of fuzzy relationsProperties of fuzzy relationsProperties of fuzzy composition
Extensions
Biblopgraphy202 241 Relations
Organizational
bull Last week (2 weeks from now) there will be a short test (max 5points) during the tutorials
bull This week we are having the last theoretical lecture
203 241 Relations
Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
204 241 Relations
Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
204 241 Relations
Residue implication
Despite the lack of a uniform definition of fuzzy implication there is auseful class of implications
Defintion
The R-implication (residuum bdquoreziduovanaacute implikaceldquo) is a functionobtained from a fuzzy T-norm
120572 1113699rArr∘ 120573 = sup120574 | 120572 and∘ 120574 le 120573 (RI)
205 241 Relations
R-implication Examples (1)
Standard implication (Goumldel) isderived from (RI) using thestandard cojunctionand1113700
120572 1113699rArr1113700120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573 otherwise
(3)
206 241 Relations
R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
207 241 Relations
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Organizational
bull Last week (2 weeks from now) there will be a short test (max 5points) during the tutorials
bull This week we are having the last theoretical lecture
203 241 Relations
Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
204 241 Relations
Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
204 241 Relations
Residue implication
Despite the lack of a uniform definition of fuzzy implication there is auseful class of implications
Defintion
The R-implication (residuum bdquoreziduovanaacute implikaceldquo) is a functionobtained from a fuzzy T-norm
120572 1113699rArr∘ 120573 = sup120574 | 120572 and∘ 120574 le 120573 (RI)
205 241 Relations
R-implication Examples (1)
Standard implication (Goumldel) isderived from (RI) using thestandard cojunctionand1113700
120572 1113699rArr1113700120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573 otherwise
(3)
206 241 Relations
R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
207 241 Relations
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
204 241 Relations
Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
204 241 Relations
Residue implication
Despite the lack of a uniform definition of fuzzy implication there is auseful class of implications
Defintion
The R-implication (residuum bdquoreziduovanaacute implikaceldquo) is a functionobtained from a fuzzy T-norm
120572 1113699rArr∘ 120573 = sup120574 | 120572 and∘ 120574 le 120573 (RI)
205 241 Relations
R-implication Examples (1)
Standard implication (Goumldel) isderived from (RI) using thestandard cojunctionand1113700
120572 1113699rArr1113700120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573 otherwise
(3)
206 241 Relations
R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
207 241 Relations
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy implication
We knownot∘ and∘ and∘or Unfortunately no nice formula for ∘rArr∘
Definition
Fuzzy implication is any function
∘rArr∘ ∶ [0 1]2 rarr [0 1] (1)
which overlaps with the boolean implication on x y isin 0 1
(x ∘rArr∘ y) = (xrArr y) (2)
204 241 Relations
Residue implication
Despite the lack of a uniform definition of fuzzy implication there is auseful class of implications
Defintion
The R-implication (residuum bdquoreziduovanaacute implikaceldquo) is a functionobtained from a fuzzy T-norm
120572 1113699rArr∘ 120573 = sup120574 | 120572 and∘ 120574 le 120573 (RI)
205 241 Relations
R-implication Examples (1)
Standard implication (Goumldel) isderived from (RI) using thestandard cojunctionand1113700
120572 1113699rArr1113700120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573 otherwise
(3)
206 241 Relations
R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
207 241 Relations
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Residue implication
Despite the lack of a uniform definition of fuzzy implication there is auseful class of implications
Defintion
The R-implication (residuum bdquoreziduovanaacute implikaceldquo) is a functionobtained from a fuzzy T-norm
120572 1113699rArr∘ 120573 = sup120574 | 120572 and∘ 120574 le 120573 (RI)
205 241 Relations
R-implication Examples (1)
Standard implication (Goumldel) isderived from (RI) using thestandard cojunctionand1113700
120572 1113699rArr1113700120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573 otherwise
(3)
206 241 Relations
R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
207 241 Relations
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
R-implication Examples (1)
Standard implication (Goumldel) isderived from (RI) using thestandard cojunctionand1113700
120572 1113699rArr1113700120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573 otherwise
(3)
206 241 Relations
R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
207 241 Relations
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
R-implication Examples (2)
Łukasiewicz implication isderived from (RI) using theŁukasiewicz cojunctionand1113693
120572 1113699rArr1113693120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 1205731 minus 120572 + 120573 otherwise
(4)
207 241 Relations
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
R-implication Examples (3)
Algebraic implication (GougenGaines) is derived from (RI) usingthe algebraic cojunctionand1113682
120572 1113699rArr1113682120573 =
⎧⎪⎨⎪⎩
1 if 120572 le 120573120573120572 otherwise
(5)
208 241 Relations
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
R-implication Properties
Theorem 209
Letand∘ be a continuous fuzzy conjunction Then R-implication satisfies
120572 1113699rArr∘ 120573 = 1 iff 120572 le 120573 (I1)
1 1113699rArr∘ 120573 = 120573 (I2)
120572 1113699rArr∘ 120573 is not increasing in 120572 and not decreasing in 120573 (I3)
209 241 Relations
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
R-implication Properties
Proof of theorem 209 Letrsquos denote 120574 | 120572 and∘ 120574 le 120573 = Γ
bull Proving (I3) uses monotonicity Increasing 120572 can only shrink Γ andincreasing 120573 can only enlarge Γ
bull Proving (I2) is easy 1 1113699rArr∘ 120573 = sup120574 | 1and∘ 120574 le 120573 From definition of
and∘ we write 11113699rArr∘ 120573 = sup120574 | 120574 le 120573 = 120573
210 241 Relations
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
R-implication Properties
Proof of theorem 209 (contd)
bull For (I1) one needs to check 2 casesbull If 120572 le 120573 then 1 isin Γ because 120572 and∘ 1 = 120572 le 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is true for all possible values of 120574
bull If 120572 gt 120573 then 1isinΓ because 120572 and∘ 1 = 120572 gt 120573 and therefore the
condition 120572 and∘ 120574 le 120573 is false for 120574 = 1
211 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
S-implication
Defintion
The S-implication is a function obtained from a fuzzy disjunction∘or
120572 1113700rArr∘ 120573 = not1113700 120572∘or 120573 (SI)
ExampleKleene-Dienes implication from 1113700or
120572 1113700rArr1113700120573 = max(1 minus 120572 120573) (6)
212 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Generalized fuzzy inclusion
Previously we used the logical negationnot∘ to define the set
complement the conjunctionand∘ to define the set intersection etc
Can we use the implication ∘rArr∘ to define the the fuzzy inclusion
Definition
The generalized fuzzy inclusion∘sube∘ is a function that assigns a degree to
the the inclusion of set A isin 120125(Δ) in set B isin 120125(Δ)
A∘sube∘ B = inf
xisin1113655A(x) ∘rArr∘ B(x) (7)
213 241 Relations
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Generalized fuzzy inclusion Example
214 241 Relations
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy inclusion (non-generalized)
Definition
The fuzzy inclusionsube is a predicate (assigns a truefalse value) whichhold for two fuzzy sets A B isin 120125(Δ) iff
120583A(x) le 120583B(x) for all x isin Δ (8)
215 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy inclusion (non-generalized)
In vertical representation the definition has a straightforwardequivalent
120583A le 120583B (9)
In horizontal representation there is a theorem
Theorem 216
Let A B isin 120125(Δ) Asube B if and only if
120449A(120572) sube 120449B(120572) for all 120572 isin [0 1] (10)
216 241 Relations
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy inclusion (non-generalized)
Proof of theorem 216
rArr Assume Asube B and x isin 120449A(120572) for some value 120572 If 120572 le A(x) thenA(x) le B(x) (from the definition of Asube B) and therefore x isin 120449B(120572)and 120449A(120572) sube 120449B(120572)
lArr Assume 120449A(120572) sube 120449B(120572) Firstly recall the horizontal-verticaltranslation formula 120583A(x) = sup120572 isin [0 1] | x isin 120449A(120572) Since120572 | x isin 120449A(120572) sube120572 | x isin 120449B(120572) the inequalityA(x) le sup120572 | x isin 120449B(120572) le B(x) holds
217 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cutworhiness
We ended up with 2 equal definitions of set inclusion using verticaland horizontal representation Can we generalize this
Cutworhiness
Let P be a predicate (returns truefalse) over fuzzy sets P is calledcutworthy (bdquořezově dědičnaacute vlastnostldquo) if the implication holds
P(A1 An) rArr P(120449A1(120572) 120449An
(120572)) for all 120572 isin [0 1] (11)
There is a related notion We define P as cut-consistent (bdquořezověkonzistentniacuteldquo) using the same definition but replacingrArr withhArr
218 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin Δ
bull Being crisp is
219 241 Relations
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cutworhiness Examples
bull The theorem 216 can be stated as ldquoSet inclusion iscut-consistentrdquo
Brain teasers
bull Strong normality of A is defined as A(x) = 1 for some x isin ΔStrong normality is cut-consistent A is strongly-normal iff everyits cut is non-empty iff every cut strongly normal
bull Being crisp iscutworthy but not cut-consistent Every cut is crisp by definitiontherefore cutworthiness But even non-crisp sets have crisp cutstherefore the property is not not cut-consistent
219 241 Relations
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Google ldquofuzzyrdquo
Sources M Taylorrsquos Weblog M Taylorrsquos Weblog Eddiersquos Trick Shop
220 241 Relations
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Google ldquoprobabilityrdquo
Sources Life123 WhatWeKnowSoFar Probability Problems
221 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy vs probability
bull Vagueness vs uncertainty
bull Fuzzy logic is functional
222 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations
Definition
A binary crisp relation R from X onto Y is a subset of the cartesianproduct X times Y
R isin ℙ(X times Y) (12)
Definition
The inverse relation R-1 to R is a relation from Y to X st
R-1 = (y x) isin Y times X | (x y) isin R (13)
223 241 Relations
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Inverse
Definition
Let XY Z be sets Then the compound of relations RsubeXtimes Y SsubeYtimes Zis the relation
R S = (x z) isin X times Z | (x y) isin R and (y z) isin S for some y (14)
224 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube R
symmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube E
transitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube R
partial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Crisp relations Properties
The identity relation onΔ is E = (x x) | x isin Δ
Then the relation RsubeΔ times Δ is called
property using logical connectives using set axioms
reflexive forallx (x x) isin R Esube Rsymmetric (x y) isin RrArr(y x) isin R R = R-1
anti-symmetric (x y) isin R and (y z) isin RrArr y = z R cap R-1 sube Etransitive (x y) isin R and (y z) isin RrArr(x z) isin R R Rsube Rpartial order reflexive transitive and anti-symmetricequivalence reflexive transitive and symmetric
225 241 Relations
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Fuzzy relations
Definition
A binary fuzzy relation R from X onto Y is a fuzzy subset on theuniverse X times Y
R isin 120125(X times Y) (15)
Definition
The fuzzy inverse relation R-1 isin 120125(Y times X) to R isin 120125(X times Y) st
R(y x) = R-1(x y) (16)
226 241 Relations
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Projection
Defintion
Let R isin 120125(X times Y) be a fuzzy binary relation The first and secondprojection of R is
R(1)(x) =S
1114004yisinY
R(x y) (17)
R(2)(y) =S
1114004xisinX
R(x y) (18)
227 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 x2 02 04 08 1 08 06 x3 04 08 1 08 04 02
R(2)(y)
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion as
228 241 Relations
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Projection Example
R y1 y2 y3 y4 y5 y6 R(1)(x)x1 01 02 04 08 1 08 1x2 02 04 08 1 08 06 1x3 04 08 1 08 04 02 1
R(2)(y) 04 08 1 1 1 08
Sometimes there is a total projection defined as
R(T) =1114004xisinX1114004yisinY
R(x y)
But we already know this notion asHeight(R)
228 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cylindrical extension
Can we reconstruct a fuzzy relation from its projections There is anunique largest relation with prescribed projections
Definition
Let A isin 120125(X) and B isin 120125(Y) be fuzzy sets The cylindrical extension(bdquocylindrickeacute rozšiacuteřeniacuteldquo bdquokarteacutezskyacute součin fuzzy mnoinldquo) is defined as
A times B(x y) = A(x) and1113700B(y) (19)
Brain teaserWhy canrsquot there be a relation Q bigger than A times B whose projectionsare Q(1) = A and Q(2) = B
229 241 Relations
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Cylindrical extension Drawing
A(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 1 x isin [1 2]3 minus x x isin [2 3]0 otherwise
B(x) =
⎧⎪⎪⎨⎪⎪⎩
x minus 3 x isin [3 4]5 minus x x isin [4 5]0 otherwise
230 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Composition of fuzzy relations
Definition
Let XY Z be crisp sets R isin 120125(X times Y) S isin 120125(Y times Z) andand∘ some fuzzy
conjunction Then the ∘ -composition (bdquo∘ -sloenaacute relaceldquo) is
R∘ S(x z) =S
1114004yisinY
R(x y) and∘ S(y z) (20)
1 For infinite domains⋁S is computed using the sup instead ofmax
2 Instead of the ldquo for some yrdquo in crisp relations the disjunctionldquofinds such a yrdquo that maximizes the conjunction
231 241 Relations
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Example of a fuzzy relation
R(x y) =⎧⎪⎨⎪⎩
x + y x y isin 11140940 1
21114097
0 otherwiseS(x y) =
⎧⎪⎨⎪⎩
x sdot y x y isin 11140940 1
21114097
0 otherwise
232 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube R
symmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy relations
Then the relation RsubeΔ times Δ is called
property using set axioms
reflexive Esube Rsymmetric R = R-1
∘-anti-symmetric R cap∘ R-1 sube E
∘-transitive R∘ Rsube R
∘-partial order reflexive ∘-transitive and ∘-anti-symmetric∘-equivalence reflexive ∘-transitive and ∘-symmetric
233 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal
bull Symmetricity Cells symmetric over the main diagonal
bull Anti-symmetricity Cells symmetric over the main diagonal
bull For S- andA-anti-symmetricity
bull For L-anti-symmetricity
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties in a finite domain
If the universeΔ is finite the relation can be written as a matrix Theirproperties are reflected in the relationrsquos matrix
bull Reflexivity Cells on the main diagonal are 1
bull Symmetricity Cells symmetric over the main diagonal are equal
bull Anti-symmetricity Cells symmetric over the main diagonal haveconjunction equal to zero
bull For S- andA-anti-symmetricity one of the elements must be zero
bull For L-anti-symmetricity their sum must be less or equal to 1
bull Transitivity More difficult (see example on the next slide)
234 241 Relations
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Example on fuzzy relation properties
LetΔ = A B CD and R isin 120125(Δ times Δ)
R A B C D
A 05 01B 02CD 02
Fill the missing cells in the table to make R
a) S-equivalenceb) A-equivalence
235 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity
236 241 Relations
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Properties of fuzzy composition
Theorem 236
Let R S and T be relations (defined over sets that ldquomake senserdquo) Thefollowing equations hold
R∘ E = R E∘ R = R (21)
(R∘ S)-1 = S-1 ∘ R
-1 (22)
R∘ (S∘ T) = (R∘ S)∘ T (23)
(R 1113700cup S)∘ T = (R∘ T)1113700cup (S∘ T) (24)
R∘ (S1113700cup T) = (R∘ S)
1113700cup (R∘ T) (25)
(21) describes the identity element (22) the inverse of composition(23) is the asociativity (24) and (25) the right- and left-distributivity236 241 Relations
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Proof of 236
Proving (21) and (22) is trivial
R∘ (S∘ T)(xw) =S
1114004y
R(x y) and∘ S∘ T(yw) (26)
=S
1114004y
R(x y) and∘
⎛⎜⎜⎝
S
1114004z
S(y z) and∘ T(zw)⎞⎟⎟⎠
(27)
=S
1114004y
S
1114004z
R(x y) and∘ S(y z) and∘ T(zw) (28)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (29)
237 241 Relations
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Proof of 236 (contd)
=S
1114004z
S
1114004y
R(x y) and∘ S(y z) and∘ T(zw) (30)
=S
1114004z
⎛⎜⎜⎝
S
1114004y
R(x y) and∘ S(y z)⎞⎟⎟⎠and∘ T(zw) (31)
=S
1114004z
R∘ S(x z) and∘ T(zw) (32)
= R∘ S∘ T(xw) (33)
Proof of (24) and (25) is similar (uses the distributivity law) onlyshorter See [Navara and Olšaacutek 2001] for details
238 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Extensions Sometimes it is useful to consider
bull a 120576-reflective relation
R(x x) ge 120576 (34)
bull aweakly reflexive relation
R(x y) le R(x x) and R(y x) le R(x x) for all x y (35)
bull Relation is 1-reflective iff reflexive
bull If a relation is reflexive then it is weakly reflexive
239 241 Relations
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Extensions Sometimes it is useful to consider
bull a non-involutive negation by refusing (N2)
not∘ not∘ 120572=120572
and adopting a weaker axiom
not∘ not∘ 0 = 1 and not∘ not∘ 1 = 0 (N0)
ExampleGoumldel negation
not1113688120572 =
⎧⎪⎨⎪⎩
1 120572 = 0
0 otherwise(36)
240 241 Relations
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
Bibliography
Navara M and Olšaacutek P (2001)Zaacuteklady fuzzy mnoinNakladatelstviacute ČVUT
241 241 Relations
- Properties of fuzzy sets
-
- Fuzzy implication and fuzzy properties
- Fuzzy set inclusion and crisp predicates
-
- Intermission Probabilistic vs fuzzy
- Binary fuzzy relations
-
- Quick revision of crisp relations
- Fuzzyfication of crisp relations
- Projection and cylindrical extension
- Composition of fuzzy relations
- Properties of fuzzy relations
- Properties of fuzzy composition
-
- Extensions
- Biblopgraphy
-
