Geometry of Fuzzy Sets

Geometry of Fuzzy SetsGeometry of Fuzzy Sets

@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 2

Sets as pointsSets as points Geometry of fuzzy sets includes

Domain X={x1,…,x2}

Range of mappings [0,1]

A:X[0,1]


Classic Power SetClassic Power Set Classic Power Set: the set of all

subsets of a classic set.

Let X={x1,x2 ,x3}

Power Set is represented by 2|X|

2|X|={, {x1}, {x2}, {x3}, {x1,x2}, {x1,x3}, {x2,x3}, X}


VerticesVertices The 8 sets correspond to 8 bit vectors

2|X|={, {x1}, {x2}, {x3}, {x1,x2}, {x1,x3}, {x2,x3}, X}

2|X|={(0,0,0),(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)}

The 8 sets are the vertices of a cube


The vertices in spaceThe vertices in space

x1

x2

x3

(0,0,0)

(1,1,1)

(1,0,1)

(1,0,0)

(1,1,0)

(0,1,0)

(0,1,1)

(0,0,1)


Fuzzy Power SetFuzzy Power Set The Fuzzy Power set is the set of all

fuzzy subsets of X={x1,x2 ,x3} It is represented by F(2|X|) A Fuzzy subset of X is a point in a cube The Fuzzy Power set is the unit

hypercube


The Fuzzy CubeThe Fuzzy Cube

x1

x2

x3

(0,0,0)

(1,1,1)

(1,0,1)

(1,0,0)

(1,1,0) (0,1,0)

(0,1,1)

(0,0,1)

A={(x1,0.5),(x2,0.3),(x3,0.7)}


Fuzzy OperationsFuzzy Operations Let X={x1,x2} and A={(x1,1/3),(x2,3/4)}

Let A´ represent the complement of A

A´={(x1,2/3),(x2,1/4)}

AA´={(x1,2/3),(x2,3/4)}

AA´={(x1,1/3),(x2,1/4)}


Fuzzy Operations in the SpaceFuzzy Operations in the Space

(0,1)

1/4

3/4

(1,1)

(1,0)

x1

x2

1/3 2/3

A

A´

AA´

AA´


Paradox at the MidpointParadox at the Midpoint Classical logic forbids the middle point

by the non-contradiction and excluded middle axioms

The Liar from Crete Let S be he is a liar, let not-S be he is

not a liar Since Snot-S and not-SS t(S)=t(not-S)=1-t(S) t(S)=0.5


Cardinality of a Fuzzy SetCardinality of a Fuzzy Set The cardinality of a fuzzy set is equal to

the sum of the membership degrees of all elements.

The cardinality is represented by |A|

n

iiA xA

1

)(||


DistanceDistance The distance dp between two sets

represented by points in the space is defined as

If p=2 the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance

p

n

i

piBiA

p xxBAd

1

|)()(|),(


Distance and CardinalityDistance and Cardinality If the point B is the empty set (the

origin)

So the cardinality of a fuzzy set is the Hamming distance to the origin

n

iiA

n

iiA

xAAd

xAd

1

1

1

1

)(||),(

0)(),(


Fuzzy CardinalityFuzzy Cardinality

(0,1)

3/4

(1,1)

(1,0)

x1

x2

1/3

A

|A|=d1(A,)


Fuzzy EntropyFuzzy Entropy How fuzzy is a fuzzy set?

Fuzzy entropy varies from 0 to 1.

Cube vertices has entropy 0.

The middle point has entropy 1.


Fuzzy Entropy GeometryFuzzy Entropy Geometry

(0,1)

3/4

(1,1)

(1,0)

x1

x2

1/3

A

a

b

),(),(

)( 1

1

far

near

AAdAAd

ba

AE


Fuzzy Operations in the SpaceFuzzy Operations in the Space

(0,1)

1/4

3/4

(1,1)

(1,0)

x1

x2

1/3 2/3

A

A´

AA´

AA´

´

´)(

AA

AAAE


Fuzzy entropy, max and minFuzzy entropy, max and min T(x,y) min(x,y) max(x,y)S(x,y) So the value of 1 for the middle point

does not hold when other T-norm is chosen.

Let A= {(x1,0.5),(x2,0.5)} E(A)=0.5/0.5=1 Let T(x,y)=x.y and C(x,y)=x+y-xy E(A)=0.25/0.75=0.333…


SubsetsSubsets Sets contain subsets.

A is a subset of B (AB) iff every element of A is an element of B.

A is a subset of B iff A belongs to the power set of B (AB iff A2B).


Subsets and implicationSubsets and implication Subsethood is equivalent to the

implication relation. Consider two propositions P and Q. A is a subset of B iff there is no element

of A that does not belong to BP Q PQ

0 0 1

0 1 1

1 0 0

1 1 1


Zadeh´s definition of SubsetsZadeh´s definition of Subsets A is a subset of B iff there is no element

of A that does not belong to B A B iff A(x) B(x) for all x

P Q PQ

0 0 1

0 1 1

1 0 0

1 1 1


Subsethood examplesSubsethood examples Consider A={(x1,1/3),(x2=1/2)} and

B={(x1,1/2),(x2=3/4)} A B, but B A

(0,1)

1/2

3/4

(1,1)

(1,0)

x1

x2

1/3 1/2

A

B


Not Fuzzy SubsethoodNot Fuzzy Subsethood The so called membership dominated

definition is not fuzzy. The fuzzy power set of B (F(2B)) is the

hyper rectangle docked at the origin of the hyper cube.

Any set is either a subset or not.


Fuzzy power set sizeFuzzy power set size F(2B) has infinity cardinality. For finite dimensional sets the size of

F(2B) is the Lebesgue measure or volume V(B)

(0,1)

1/2

3/4

(1,1)

(1,0)

x1

x2

1/3 1/2

A

B

n

iiB xBV

1

)()(


Fuzzy SubsethoodFuzzy Subsethood Let S(A,B)=Degree(A B)=F(2B)(A) Suppose only element j violates

A(xj)B(xj), so A is not totally subset of B.

Counting violations and their magnitudes shows the degree of subsethood.


Fuzzy SubsethoodFuzzy Subsethood Supersethood(A,B)=1-S(A,B) Sum all violations=max(0,A(xj)-B(xj)) 0S(A,B)1

A

xxBAS

A

xxBAodSupersetho

XxBA

XxBA

))()(,0max(1),(

))()(,0max(),(


Subsethood measuresSubsethood measures Consider A={(x1,0.5),(x2=0.5)} and

B={(x1,0.25),(x2=0.9)}

6.0),(5.05.0

))5.09.0(,0max())5.025.0(,0max(1),(

75.0),(

5.05.09.05.0,0max25.05.0,0max

1),(

ABS

ABS

BAS

BAS

Geometry of Fuzzy Sets

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