operations on fuzzy sets one should not increase, beyond what is necessary, the number of entities...

Operations on Fuzzy Sets

One should not increase, beyond what is necessary, the number of entities required to explain anything.

Occam's Razor

Adriano Cruz ©2002NCE e IM/UFRJ

[email protected]

@2005 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 2

Summary

Zadeh’s Operations T-Norms S-Norms Properties of Fuzzy Sets Fuzzy Measures


Zadeh’s Definitions

Lofty Zadeh put forward the basic set operations is his seminal paper “Fuzzy Sets”, Information and Control, 1965

These operations reduce to the boolean

operations when crisp sets are used.


Union

Union: The union the two sets A and B (AB) can be defined by the membership function U(x)

(x)=max((x),(x)), x X


Intersection

Intersection: the intersection of two sets A and B (AB) can be defined by the membership function (x)

(x)=min((x),(x)), x X


Complement of a Fuzzy Set

Complement: the complement of a fuzzy set A can be defined by the membership function C(x)

μC x =1−μ A x ,x∈X

1−μ A x μ A x


Why these operators?

The crisp set operators are very well defined and understood, however when fuzzy sets are considered this definition is fuzzy and many other operations can be considered.

Fuzzy set operators must obey a set of rules that generalize the operations. The so called T-norms (T(x,y)) and the T-conorms or S-norms (S(x,y)).

T-norms generalize the and operator and t-conorms generalize the or operator


T- Norms


Intersection operation

Any t-norm operator, denoted as t(x,y) must satisfy five axioms.

T-norms map from [0,1]x[0,1] [0,1]

Let A(x), B(x), C(x) and D(x) four functions (sets). In order to simplify the notation we will use the letters a, b, c e d to represent them.


T-norms

T.1 T(0,0) = 0

T.2 T(a,b) = T(b,a) commutative

T.3 T(a,1) = a neuter

T.4 T(T(a,b),c)=T(a,T(b,c)) associative

T.5 T(c,d) <=T(a,b) if c<=a and d<=b monotonic


T-norms: comments

It can be proved that the minimum operation is a t-norm

The product operator is also a t-norm Obviously there are other operations that

satisfy these axioms It can be proved that for any t-norm

(x), (x)) <= min((x), (x))


Minimum, T-norm?

T.1 min(0,0) = 0

T.2 min(a,b) = min(b,a)

T.3 min(a,1) = a

T.4 min(min(a,b),c) = min(a,min(b,c)) = min(a,b,c)

T.5 min(c,d) <= min(a,b) if c <= a and d <= b


Union

Any s-norm operator denoted as s(x,y) must satisfy five axioms

S-norms map [0,1]x[0,1] [0,1]

Let A(x), B(x), C(x) e D(x) four fuzzy sets. In order to simplify the notation we will use the letters a,b,c and d.


S-norms or T-conorms

S.1 S(1,1) = 1

S.2 S(a,b) = S(b,a) commutative

S.3 S(a,0) = a neuter

S.4 S(S(a,b),c)=S(a,S(b,c)) associative

S.5 S(c,d) <=S(a,b) if c<=a and d<=b monotonic


S-norms: comments 1

It can be proved that the maximum is a s-norm

Obviously there are other operations that satisfy these axioms.

The addition operation do not satisfy the S.1 axiom, so it can not be used.

It can be proved that for any S-norm we have

S(x), (x)) >= max((x), (x))


S-norms: comments 2

Note that it is notit is not required the S-norm to be idempotentidempotent, that is S(a,a)=a, therefore the union of a set to itself it union of a set to itself it is not required to be equal to itselfis not required to be equal to itself.

Nor is required that the S-norm to be continuous


Maximum, S-norm?

S.1 max(0,0) = 0

S.2 max(a,b) = max(b,a)

S.3 max(a,1) = a

S.4 max(max(a,b),c) = max(a,max(b,c)) = max(a,b,c)

S.5 max(c,d) <= max(a,b) if c <= a and d <= b


Algebraic sum, S-norm?

μ A∪B a,b =a+b−a×b

S .1 11−1×1=1S . 2 a+b−a×b=b+a−b×a comutativaS . 3 a+ 0−a×0 =a


Algebraic Sum, S-norm?

S.4 S((a+b),c) = (a+b-ab) + c – (a+b-ab)c = a+b-ab+c-ac-bc+abc =a+(b+c-bc)-a(b+c-bc) = S(a,(b+c)) =a + b + c – ab – ac – bc + abc

S.5 if c <= a, d <= b, a, b, c, d <= 1a + b -ab >= c + d -cd


Other examples

Prove that T(a,b)<= min(a,b)


Other examples

• Prove that T(a,b) <= min(a,b)• T5: T(a,b) <= T(a,1) = a• T2: T(a,b) = T(b,a)• T5: T(b,a) <= T(b,1) = b• T2: T(a,b) <= min(a,b)


Pairs of T-norms and S-norms

T-norm - Drastic Product:

S-norm - Drastic Sum:

DPx,y ={min x,y if max x,y =10 x,y1 }

DS x,y ={max x,y if min x,y =01 x,y> 0 }



T-norm - Bounded Difference:

S-norm - Bounded Sum:

BD x,y =max 0, x+y−1

BS x,y =min 1, x+y



T-norm – Einstein Product:

S-norm - Einstein Sum:

EP x,y =xy

2−[ x+y− xy ]

ES x,y =x+y

1 +x . y



T-norm – Algebraic Product:

S-norm - Algebraic Sum:

AP x,y =xy

AS x,y =x+y−xy



T-norm – Hamacher Product:

S-norm - Hamacher Sum:

HP x,y =xy

x+y− xy

HS x,y =x+y−2 xy

1− xy


Four T-norm operators


Four T-conorm operators



T-norm – Dubois-Prade:

S-norm – Dubois-Prade:

DPrT x,y =xy

max p,x,y

DPrS x,y =x+y− xy−min 1− p,x,y max p ,1− x ,1− y

Obs. p is a parameter that ranges from 0 to 1.


Dubois-Prade Operators

When p=1 – Dubois-Prade T-norm becomes the

Algebraic Product (xy)– Dubois-Prade S-norm becomes the

Algebraic Sum (x+y-xy) When p=0

– Dubois-Prade T-norm becomes the min(xy)

– Dubois-Prade S-norm becomes the max(xy)



T-norm – Yager:

S-norm – Yager:

Y T x,y =1−min 1, [ 1− x p1− y p ]1 / p

Y T x,y =min 1, [ x p +y p ]1 / p

Obs. p is a parameter that ranges from 0 to .


Yager Operators

When p=1.0 – Yager T-norm becomes the bounded

difference (max(0,x+y-1))– Yager S-norm becomes the bounded sum

(min(1,x+y))

When p->– Yager T-norm converges to min(x,y)– Yager S-norm converges to max(x,y)


Complement


Fuzzy Complement axioms

A fuzzy complement operator is a A fuzzy complement operator is a continuous function continuous function NN:[0,1]:[0,1][0,1] [0,1] which meets the following axioms:which meets the following axioms:

NN(0)=1 and (0)=1 and NN(1) = 0 (boundary) (a.1)(1) = 0 (boundary) (a.1) NN((aa))NN((bb) if ) if aa bb (monotonicity) (a.2) (monotonicity) (a.2) Another Another optionaloptional requirements are requirements are

– N(x) is continuous (a.3)N(x) is continuous (a.3)– NN((NN((aa))=))=aa (involution) (a.4) (involution) (a.4)


Fuzzy Complements

All functions a.1 e a.2

Classical

Involutive a.4Continuous a.3


Usual Fuzzy Complement

Consider - Consider - NN((xx)=1-)=1-xx

NN(0)=1(0)=1 and and NN(1) = 0(1) = 0 (boundary) (boundary)

NN((aa))NN((bb)) if if aa bb (monotonicity) (monotonicity)

NN((NN((xx))=1-(1-))=1-(1-xx)=)=xx

Continuous in the intervalContinuous in the interval


Another Ex of Complement

Consider Consider N x ={1 for a≤ t0 a>t }

t,a∈[ 0,1 ]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

N (

x)

Example of complement


Another Ex of Complement 1

ConsiderConsider

Satisfy only the axiomatic requirements:Satisfy only the axiomatic requirements: NN(1)=0, N(0)=1(1)=0, N(0)=1 NN((xx) is monotonic) is monotonic NN((xx) is not continuous) is not continuous NN((xx) is not involutive) is not involutive

N x ={1 for a≤ t0 a>t }

t,a∈[ 0,1 ]


Sugeno’s complement

The operator is defined as

where s is a parameter greater than –1. For each s, we obtain a particular

complement

N s a =1−a1 +sa


Sugeno’s complement


Yager’s complement

The operator is defined as

where y is a positive parameter.

N y a = 1−a y 1 /y


Yager’s complement


Complement equilibrium

The point of equilibrium is any x for which N(x)=x

For a classical fuzzy set – x=1-x– x = 0.5

Equilibrium can be used to measure fuzziness


Properties


Fuzzy Sets Properties

Comutativity A∪B=B∪AA∩B=B∩A

Associativity A∪ B∪C = A∪B ∪CA∩ B∩C = A∩B ∩C

Distributivity A∪ B∩C = A∪B ∩ A∪C A∩ B∪C = A∩B ∪ A∩C

Absorption A∩ A∪B =AA∪ A∩B =A


Fuzzy Sets Properties

Identity A∪∅=AA∩∅=∅A∪X=XA∩X=A

De Morgan A∩B= A∪BA∪B= A∩B


Checking Properties

min a,b = {a if a≤bb if b<a }

min a,b =a+b−∣a−b∣

2

Remember that

and

max a,b = {b if a≤ba if b<a }

max a,b =a+b+∣a−b∣

2


Checking Properties

Lets check the absorption property

A∪ A∩B =Amax [ μ A x ,min μ A x ,μB x ]=μ A x

A∩B ¿a+b−∣a−b∣

2

A∪ A∩B ¿a+

a+b−∣a−b∣2

∣a−a+b−∣a−b∣

2∣

2


Checking Properties

A∪ A∩B ¿

3a+b−∣a−b∣2

∣a−b+∣a−b∣

2∣

2

¿3a +b−∣a−b∣∣a−b+∣a−b∣∣

4

If a≥b

A∪ A∩B ¿3a +b− a−b a−b a−b

4A∪ A∩B ¿ a


Checking Properties

A∪ A∩B ¿3a +b−∣a−b∣∣a−b+∣a−b∣∣

4if a<b

A∪ A∩B ¿3a +b+a−b ∣a−b −a−b ∣

4A∪ A∩B ¿ a


Laws of Aristotle

Law of Non-ContradictionLaw of Non-Contradiction: “One cannot say of something that it is and that it is not in the same respect and at the same time”.

One element must belong to a set or its complement.

Since the intersection between one set and its complement may be not empty we may have

A∩A≠∅


Law of non-contradiction

adultsNon adults

adults∩adults

adults∩adults

1.0

1.0

0.5


Laws of Aristotle

Law of excluded middleLaw of excluded middle: for any proposition P, it is true that (P or not-P).

So the union of a set and its complement should give all the universe.

However the result may be not the universe

A∪A≠X


Law of non-contradiction

adultsNon adults1.0

adults∪adults

1.0


Measures


Fuzzy Entropy

The entropy of a fuzzy set is defined as

c is a counting operation (addition or integration) defined over the set.

Note that for a crisp set the numerator is always 0 and the entropy of a crisp set is always 0.

E A =c A∩ A c A∪ A


Fuzzy Entropy

adultsNo adults

adult∩adult

The entropy of the adult fuzzy set is

1

10 20 30

c A∩A 5c A∪A 2020−5 35

E A 5

350 .14


Fuzziness Measurements

A measure of fuzziness is a function

P(X) is the set of all fuzzy subsets of X

There are three requirements that a meaningful measure must satisfy

Only one is unique; the other depend on the meaning of fuzziness

f : Ρ X ℜ


Requirements

F1: f(A) = 0 iff A is a crisp set

F2:

F3: f(A) assumes the maximum value iff

A is maximally fuzzy.

A≺B means A is less fuzzy than B

if A≺B then f A ≤ f B


Measurement Based on Distance

One measure of fuzziness is defined in terms of a metric distance from the set A to the nearest crisp setnearest crisp set.

Distance from point A(a0,a1,…,an) to B(b0,b1,…,bn)

d p A,B =p∑i=1

n

∣a i−b i∣p


Measurement Based on Distance

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

d p A,B =p∑i=1

n

∣a i−b i∣p


The Geometry of Sets

Crisp Sets can be view as points in a space

Fuzzy sets are also part of the same space

Using these concepts it is possible to measure distances from crisp to fuzzy sets.


Classic Power Set

Classic Power Set: the set of all subsets of a classic set.

Let consider X={x1,x2 ,x3}

Power Set is represented by 2|X|

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


Vertices

The 8 sets can correspond to 8 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 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 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 Cube

x1

x2

x3

(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)}

0.3

0.5

0.7


Fuzzy 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 Space

(0,1)

1/4

3/4

(1,1)

(1,0)

x1

x2

1/3 2/3

A(1/3,3/4)

A´(2/3,1/4)

AA´

AA´


Paradox 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 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|

∣A∣=∑i=1

n

μ A x i


Distance

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

d p A,B =p∑i=1

n

∣μ A x i − μB x i ∣p


Distance 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

d1 A,φ =∑i=1

n

∣μ A x i −0∣

d1 A,φ =∣A∣=∑i= 1

n

∣μ A x i ∣


Fuzzy Cardinality

(0,1)

3/4

(1,1)

(1,0)

x1

x2

1/3

A

|A|=d1(A,)


Fuzzy 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 Operations in the Space

(0,1)

1/4

3/4

(1,1)

(1,0)

x1

x2

1/3 2/3

A

A´

AA´

AA´

E A =∣A∩ A´∣∣A∪ A´∣


Fuzzy Entropy Geometry

(0,1)

3/4

(1,1)

(1,0)

x1

x2

1/3

A

a

b

E A=ab=

d1 A,Anear

d1 A,A far


Fuzzy 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…


Subsets

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 A2|B|).


Subsethood 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 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 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

V B =∏i=1

n

μ B x i


Fuzzy 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 Subsethood

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

Supersethood A,B =∑x∈X

max 0, μ A x −μ B x

∣A∣

S A,B =1−∑x∈X

max 0, μ A x −μ B x

∣A∣


Subsethood measures

Consider A={(x1,0.5),(x2=0.5)} and B={(x1,0.25),(x2=0.9)}

S A,B =1−max 0, 0 . 5−0 . 25 max 0, 0 . 5−0 . 9

0 . 50 . 5 S A,B =0 . 75

S B,A=1−max 0,0 . 25−0 . 5 max 0,0 . 9−0 .5

0 . 50 . 5S B,A=0 . 6

operations on fuzzy sets one should not increase, beyond what is necessary, the number of entities...

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