Fuzzy Relations Hand Out

7/26/2019 Fuzzy Relations Hand Out

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Fuzzy RelationsFuzzy Relations

Adriano Joaquim de O Cruz

2009

NCE/[email protected]

@2009 Adriano Cruz NCE e IM - UFRJ Relations 2

Summary

Introduction

Crisp Relations

Operations with Crisp Relations

Fuzzy Relations

Operations with Fuzzy Relations


Introduction

Relations are associations betweenelements of two or more sets.

If the degree of association is one orzero there is a crisp association.

Degrees of association can be between0 and 1 in a fuzzy relation

For example the relation x is greaterthan y.


Functions and Relations

Functions and Relations are mappings.

Functions are many to one mappings.

Relations can map many to many.


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Functions

XXYYf(X)f(X)


Relations

XXYYf(X)f(X)


Cartesian Product

The Cartesian product of two crisp setsXand Yis defined as

For nsets (Xi) the Cartesian product isdefined as

}|),{( YyeXxyxYX =

}..1,|),,,{( 2121 niXxxxxXXX iinn == KL


Crisp Relations

An relation is a subset of the Cartesianproduct

The Cartesian product can be considered arelation without restrictions.

A relation is also a set, therefore the basicset concepts such as union, intersection,complement, can be applied.

nn XXXXXXR KK 2121 ),,,(


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Crisp Relations


Characteristic Function

Shows the strength of the relationbetween the pairs.

Every tuple that belongs to the relation

receives a value 1 and 0 otherwise.

=

Ryx

RyxyxR

),(0

),(1),(


Binary Relations

A relation between two sets Xand Yis

called a binary relation (R(X,Y)).

Binary relations can be defined on a

single set (R(X,X)).

These relations are often referred as

directed graphs or digraphs


Representing Relations

Sets of ordered tuples.

Consider a family and the relation is cousin of

XXR

RMarcoDboraClaraBeatrizX

=

ofcousin},,,{

)},(),,(),,(

),,(),,(),,(

),,(),,{(

ClaraMarcoBeatrizMarcoClaraDbora

BeatrizDboraMarcoClaraDboraClara

MarcoBeatrizDboraBeatrizR =


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N-dimensional membership matrices

0011

0011

1100

1100

Cousin

Marco

Dbora

Clara

Beatriz

MarcoDboraClaraBeatriz



Diagrams that display elements as pointsand the relations as arrows betweenpoints (Sagittal diagrams).

Beatriz

Clara

Dbora

Marco

Beatriz

Clara

Dbora

Marco



Simple Diagrams.

Beatriz

Clara

Dbora

Marco



Equations

yRxXyx ,


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Special Relations

Consider a set A={0,1,2} and the

relations shown below on A A

Identity Relation I = {0,0),(1,1),(2,2)}

Universal Relation

U={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0)

,(2,1),(2,2)}


Relations - Continuous Universes

xy 2=

y


Properties of Crisp Relations

Symmetric: Ris symmetric if and only if(x,y)R e (y,x)R for all element xX e yY.

Asymmetric: R isasymmetric if there is noelements xX and yY such as (x,y)Rand(y,x)R.

Antisymmetric: R is antisymmetric if for all

xX and yY, whenever (x,y)R and (y,x)Rthen x=y.


Examples of Symmetric Relations

Is odd and is odd too Is married to Is equal to

325

4

313

2

211

54321

x

y

x is odd and y is odd too


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Transitive: R is transitive if for all x,y,z X, if (x,y)R and (y,z) R then (x,z)R.

Antitransitive: R is antitransitive if (x,z) R, whenever (x,y)R and (y,z) R.

Connected: R isconnected if for all x,y

X, if xythen (x,y)R or (y,x)R.


Examples of Transitive Relations

Is greater than Is subset of Divisibility Implies

x

y

x is divisible by y 8

7

6

5

4

3

2

1

87654321


Transitive Relations

The converse of a transitive relation is alwaystransitive: if is a subset of is transitive and is asuperset of is its converse then is a superset of istransitive.

The intersection of two transitive relations is alwaystransitive: if "was born before" and "has the samefirst name as" are transitive then "was born beforeand has the same first name as" is transitive.

The union of two transitive relations is not alwaystransitive. For instance "was born before or has thesame first name as" is not generally a transitiverelation.

The complement of a transitive relation is not alwaystransitive.



Left Unique: R is left unique when for allx,y,zX, if (x,z)R and (y,z)Rthen x=y.

Right Unique: Ris unique when for allx,y,zX, if (x,y)Rand (x,z)Rthen y=z.

Biunique: a relation Rwhich is both leftunique and right unique is called biunique.


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Example Relation = cousin of

The relation is not reflexive because no one iscousin of himself, therefore it is antireflexiveand irreflexive.

The relation is symmetric because if Beatriz iscousin of Dbora then Dbora is cousin of

Beatriz.Therefore cousin ofis not asymmetric.

Beatriz

Clara

Dbora

Marco



The relation is also not antisymmetricbecause it is not reflexive norasymmetric.

Beatriz

Clara

Dbora

Marco



The relation is not transitive becauseDbora is Claras cousin and Clara isMarcos cousin, but Dbora is not acousin of Marco.

Beatriz

Clara

Dbora

Marco



The relation is not connected becausethere are pairs of different elements towhich the relation is not applicable. Forexample Marco is no cousin of Dbora.

Beatriz

Clara

Dbora

Marco


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The relation is not left unique becauseBeatriz and Clara are different persons andboth are Dboras cousin.

Beatriz

Clara

Dbora

Marco



The relation is not right unique becauseBeatriz is a cousin of Dbora and Marcowhich are different persons.

The relation is neither left unique nor right

unique therefore it is not biunique.

Beatriz

Clara

Dbora

Marco


Crisp Equivalence Relations

A crisp binary relation R(X,X) that is reflexive,symmetric and transitive is called anequivalence relation.

The similarity of triangles is an equivalence

relation.

Work at the same building is an equivalence

relation.


Crisp Equivalence Relations

For each element xX, there is a crisp setAx, which contains all elements of Xthat arerelated to xby the equivalence relation R.

Ax= { y| (x,y) R(x,y) } xR due to reflexivity of R. Each member of Ax is related to all the other

members of Axbecause Ris transitive andsymmetric.

No element of Ax is related to any element ofXnot included in Ax


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Crisp Equivalence Relations Ex

Let X= {1,2,3,,10}

Let R(X,X) = {(x,y) | x% 3 y% 3}, % isremainder when divided by 3

This relation is reflexive, symmetric andtransitive therefore is an equivalence relationon X.

The three equivalence classes are:

A1 = A4 = A7 = A10 = {1,4,7,10}

A2= A5 = A8 = {2,5,8} A3= A6 = A9 = {3,6,9}


Crisp Tolerance Relations

Relations that are reflexive and symmetric arecalled a compatibility relations or tolerancerelations.

The relation city xis close to city y is atolerance relation.

Lisbon is obviously close to itself (reflexive).

If Lisbon is close to Paris then Paris is close toLisbon (symmetric).

It is not certain that if Lisbon is close to Paris

and Paris is close to Berlin then Lisbon is closeto Berlin (not transitive).


Partial Order

A Relation that is reflexive, antisymmetric andtransitive is called a partial order relations ( ou).

The relation A is a subet of B (A B) is a partial

order. A A (reflexive) If A B and B C then A C (transitive) If A B and B A then A = B

(antisymetric) A Partial Order does not guarantee that all

pairs are comparable


Strict Order

A Relation that is antireflexive,antisymmetric and transitive is called astrict order relation (< ou >).


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Types of Binary Relations

Reflexive Antireflex Symmet Antisymm Transitive

Equiv X X X

QuasiEquiv

X X

Tolerance X X

PartialOrder

X X X

StrictOrder

X X X



Let Rand Sbe two relations on the

Cartesian product XY.

Let Oand Ibe

=

000

000

000

L

MOMM

L

L

O

=

111

111

111

L

MOMM

L

L

E



Operations

Crisp Relations are basically setsdefined over higher-dimensionaluniverses, that is Cartesian products

Usual operations such as union,intersection and so on are alsoapplicable.


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Let Rand Sbe two relations on the

Cartesian product XY.

Let Oand Ebe

=

000

000

000

L

MOMM

L

L

O

=

111

111

111

L

MOMM

L

L

E


Properties of Crisp Operations

),(1),(

:

)],(),,(min[),(

:

)],(),,(max[),(

:

yxyx

oComplement

yxyxyxSR

Interseo

yxyxyxSR

Unio

RR

SRSR

SRSR

=

=

=



)()()(

)()()(

)()(

)()(

CABACBA

CABACBAvityDistributi

CBACBACBACBAityAssociativ

ABBA

ABBAtyComutativi

=

=

=

=

=

=



AXA

XXA

AAAIdentity

AAA

AAAyIdempotenc

=

=

=

=

=

=


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=

=

AAmiddletheof

EAAExclusion

BABA

BABAMorganDe

=

=


Composition of Crisp Relations

X Y Z

R S

T=RS


Composition of Crisp Relations

=YX

SRy

yxyxSR )],(),([ o

productor ==

=

min

max

The operation is similar to amatrix multiplication.


Example of Composition

x1

x2

x3

x4

y1

y2

y3

z1

z2

z3

ZYS

YXR

=

=


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=

100

100

010

011

R

=

001

100

001

S

=

001001

100

101

SR o



x1

x2

x3

x4

z1

z2

z3

=

001

001

100

101

SR o

Fuzzy Relations


Fuzzy Relations

Fuzzy relations (R) map elements froma set (X) into a set (Y).

The strength of the relations is given bymembership functions that can varybetween 0 and 1.

R:XY[0:1]


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Fuzzy Relations

Let Aibe fuzzy sets.

A fuzzy relation is a subset of the Cartesianproduct

The Cartesian product can be considered anrelation without restrictions.

nn AAAAAAR KK 2121 ),,,(


Properties of Fuzzy Relations

Let Xand Ytwo fuzzy subsets defined on anUniverse U.

Let the elements x Xand y Ywithmembership degrees X(x) e Y(y).

Let Sbe the Cartesian product X Y.

Let Rbe a fuzzy relation on S.



Properties with similar definitions to crisprelations:

Reflexive - R(x,x) = 1

Irreflexive - R(x,x) 1 for some x

Antireflexive - R(x,x) 1 for all x

-Reflexive - R(x,x) >=




Symmetric - R(x,y) = R(y,x)

Assymetric - R(x,y) R(y,x) for some x,yX

Antisymetric when R(x,y) > 0 and

R(y,x)>0 implies that x= yfor all x,y X


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Connected

Left unique, right unique, biunique



Transitive: Ris transitive if for all x,y,zwe

have that if (x,y)R and (y,z) R then(x,z)R.

),min(),(then

),(and),(If

21

21

==

kiR

kjRjiR

xx

xxxx


Fuzzy Similarity Relations

A fuzzy binary relation that is reflexive,symmetric and transitive is known as asimilarity relation.

An equivalence relation groupselements that are equivalent.

The similarity can be viewed from twodifferent points of view.



The similarity can be considered togroup elements into crisp sets whosemembers are similar to each other tosome degree.

When the degree is equal to one thegrouping is an equivalence class.


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The similarity can also consider thedegree of similarity that the elements ofXhave to some specific element xX.

Then for each Xa similarity class canbe defined.


Fuzzy Similarity Relations Ex

1.5.50.500g

.51.90.900f

.5.910100e

00010.4.4d

.5.910100c

000.401.8b

000.40.81a

gfedcba



X = {a, b, c, d, e, f, g}

Level set = { 0, .4, .5, .8, .9, 1}

Five nested partitions



a b d c e f g=.4

a b d c e f g=.5

a b d c e f g=.8

a b d c e f g=.9

a b d c e f g=1


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Fuzzy Tolerance Relations

Relations that are reflexive and symmetricare called tolerance relations.

The fuzzy relation The city xis close tothe city y is a tolerance relation.

Orderings


Ordering Characteristics

Similarity and tolerance arecharacterized by symmetry.

Ordering relations require asymmetry(or antisymmetry) and transitivity.


Partial Ordering

A crisp binary relation R(X,X) that is reflexive,antisymmetric and transitive is called a partialordering.

The symbol is suggestive of the propertiesof this relation

x ydenotes (x,y) Rand xprecedes y

A partial ordering does not guarantee(antisymmetric x y, but ymay not x) thatall pairs of elements x, y in X are comparable(x yor y x)


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Strict Ordering

A crisp binary relation R(X,X) that isantireflexive, antisymmetric andtransitive is called a strict ordering.

Therefore (x,x) R,




Let Rand Sbe two fuzzy relations on

the Cartesian Product XY.

Let the relations

=

000

000

000

L

MOMM

L

L

O

=

111

111

111

L

MOMM

L

L

E



),(1),(

:

)],(),,(min[),(

:

)],(),,(max[),(

:

yxyx

Complement

yxyxyxSR

onIntersecti

yxyxyxSR

Union

RR

SRSR

SRSR

=

=

=


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Properties of Operations

)()()(

)()()(

)()(

)()(

CABACBA

CABACBAvityDistributi

CBACBA

CBACBAityAssociativ

ABBA

ABBAtyComutativi

=

=

=

=

=

=



AXA

XXA

A

AAIdentity

AAA

AAAeIdempotenc

=

=

=

=

=

=



AAMiddleof

EAAExclusion

BABA

BABAMorganDe

=

=


Composition of Fuzzy Relations

X Y Z

R S

T=RS


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Composition of Fuzzy Relations

=YX

SRy

yxyxSR )],(),([ o

productor ==

=

min

max

The operation is similar to a

matrix multiplication.


Example of Fuzzy Composition

x1

x2

x3

x4

y1

y2

y3

z1

z2

z3

ZYS

YXR

o

o

=

=

1.0

0.8

0.9

0.8

1.0

0.9

0.8

0.7



=

0.100

8.000

09.00

08.01

R

=

007.0

8.000

009.0

S

=

0000007.000

0000007.000

08.00000000

08.00000009.0

SR o

)]7.00()08.0()9.01[(),( 11 =zxR



x1

x2

x3

x4

z1

z2

z3

=

007.0

007.0

8.000

8.009.0

SR o

0.9

0.8

0.80.7

0.7


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Relation Example


Relation Definition

Consider a relation that express therelation petite in terms of height andweight of a female

Consider the range of these variablesas

Height = [1.51, 1.54, 1.57, , 1.69]

Weight = [40.8, 43.1, 45.4, 47.6, 49.9,52.2, 54.4, 56.7]


Relation Matrix

000000001.69

000000.40.40.61.66

00000.20.40.60.81.63

000.30.511111.60

00.10.7111111.57

0.10.30.9111111.54

0.20.51111111.51

56.754.452.249.947.645.443.140.8


Questions?

What is the degreethat a female with aspecific height and a specific weight isconsidered to be petit?

Relation is equivalent to the membership functionof a multidimensional fuzzy set.

What is the possibilitythat a petit person hasa specific of height and weight measures?

Relation is the possibility distribution assigned toa petit person whose height and weight areunknown.


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Other question?

Given a two-dimensional relation andthe possible values of one variable,infer the possible values of the othervariable.

What is the possible weightof a personwho is about 1.63?

Is it possible to this person to weigh49.9?


Answering the question

What is the possibility that the person is 1.51given that she is about 1.63?

What is the possibility that the person is 1.51and weighs 49.9?

If both answers are positive then 49.9 is a

possible weigh.

Continue and ask: what is the possibility thatthe person is 1.54?

What is the possibility that she weighs 49.9?


Rule of inference

(Possible-height(1.51) Petite (1.51,40.8))


(Possible-height(1.69) Petite (1.69,40.8)) Possible-weight(40.8)



Rule of inference

= Petite jixHeight ixweight hj whhw i),()()(

)()(

Composition max-min Composition max-prod

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