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 ©2002 NCE e IM/UFRJ

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 2 Summary Zadeh’s Operations T-Norms S-Norms Properties of Fuzzy Sets Fuzzy Measures

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 3 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.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 4 Union Union: The union the two sets A and B (A  B) can be defined by the membership function  U (x)   (x)=max(   (x),   (x)), x  X

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 5 Intersection Intersection: the intersection of two sets A and B (A  B) can be defined by the membership function   (x)   (x)=min(   (x),   (x)), x  X

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 6 Complement of a Fuzzy Set Complement: the complement of a fuzzy set A can be defined by the membership function  C (x)

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 7 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 8 T- Norms

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 9 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.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 10 T-norms T.1T(0,0) = 0 T.2 T(a,b) = T(b,a)commutative T.3 T(a,1) = aneuter 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 11 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))

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 12 Minimum, T-norm? T.1min(0,0) = 0 T.2min(a,b) = min(b,a) T.3min(a,1) = a T.4 min(min(a,b),c) = min(a,min(b,c)) = min(a,b,c) T.5min(c,d) <= min(a,b) if c <= a and d <= b

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 13 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.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 14 S-norms or T-conorms S.1S(1,1) = 1 S.2 S(a,b) = S(b,a)commutative S.3 S(a,0) = aneuter 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 15 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))

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 16 S-norms: comments2 it is not idempotent union of a set to itself it is not required to be equal to itself Note that it is not required the S-norm to be idempotent, that is S(a,a)=a, therefore the union of a set to itself it is not required to be equal to itself. Nor is required that the S-norm to be continuous

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 17 Maximum, S-norm? S.1max(0,0) = 0 S.2max(a,b) = max(b,a) S.3max(a,1) = a S.4 max(max(a,b),c) = max(a,max(b,c)) = max(a,b,c) S.5max(c,d) <= max(a,b) if c <= a and d <= b

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 18 Algebraic sum, S-norm?

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 19 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 <= 1 a + b -ab >= c + d -cd

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 20 Other examples Prove that T(a,b)<= min(a,b)

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 21 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)

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 22 Pairs of T-norms and S-norms T-norm - Drastic Product: S-norm - Drastic Sum:

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 23 Pairs of T-norms and S-norms T-norm - Bounded Difference: S-norm - Bounded Sum:

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 24 Pairs of T-norms and S-norms T-norm – Einstein Product: S-norm - Einstein Sum:

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 25 Pairs of T-norms and S-norms T-norm – Algebraic Product: S-norm - Algebraic Sum:

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 26 Pairs of T-norms and S-norms T-norm – Hamacher Product: S-norm - Hamacher Sum:

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 27 Four T-norm operators

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 28 Four T-conorm operators

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 29 Pairs of T-norms and S-norms T-norm – Dubois-Prade: S-norm – Dubois-Prade: Obs. p is a parameter that ranges from 0 to 1.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 30 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)

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 31 Pairs of T-norms and S-norms T-norm – Yager: S-norm – Yager: Obs. p is a parameter that ranges from 0 to .

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 32 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)

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 33 Complement

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 34 Fuzzy Complement axioms A fuzzy complement operator is a continuous function N:[0,1]  [0,1] which meets the following axioms: A fuzzy complement operator is a continuous function N:[0,1]  [0,1] which meets the following axioms: N(0)=1 and N(1) = 0 (boundary) (a.1) N(0)=1 and N(1) = 0 (boundary) (a.1) N(a)  N(b) if a  b (monotonicity) (a.2) N(a)  N(b) if a  b (monotonicity) (a.2) Another optional requirements are Another optional requirements are –N(x) is continuous (a.3) –N(N(a))=a (involution) (a.4)

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 35 Fuzzy Complements All functions a.1 e a.2 Classical Involutive a.4 Continuous a.3

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 36 Usual Fuzzy Complement Consider - N(x)=1-x Consider - N(x)=1-x N(0)=1 and N(1) = 0 (boundary) N(0)=1 and N(1) = 0 (boundary) N(a)  N(b) if a  b (monotonicity) N(a)  N(b) if a  b (monotonicity) N(N(x))=1-(1-x)=x N(N(x))=1-(1-x)=x Continuous in the interval Continuous in the interval

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 37 Another Ex of Complement Consider Consider

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 38 Another Ex of Complement 1 Consider Consider Satisfy only the axiomatic requirements: Satisfy only the axiomatic requirements: N(1)=0, N(0)=1 N(1)=0, N(0)=1 N(x) is monotonic N(x) is monotonic N(x) is not continuous N(x) is not continuous N(x) is not involutive N(x) is not involutive

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 39 Sugeno’s complement The operator is defined as where s is a parameter greater than –1. For each s, we obtain a particular complement

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 40 Sugeno’s complement

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 41 Yager’s complement The operator is defined as where y is a positive parameter.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 42 Yager’s complement

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 43 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 44 Properties

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 45 Fuzzy Sets Properties

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 46 Fuzzy Sets Properties

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 47 Checking Properties Remember that and

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 48 Checking Properties Lets check the absorption property

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 49 Checking Properties

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 50 Checking Properties

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 51 Laws of Aristotle Law of Non-Contradiction Law 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 52 Law of non-contradiction adultsNon adults

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 53 Laws of Aristotle Law of excluded middle Law 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 54 Law of non-contradiction adultsNon adults 1.0

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 55 Measures

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 56 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.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 57 Fuzzy Entropy adultsNo adults The entropy of the adult fuzzy set is

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 58 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 59 Requirements F1: f(A) = 0 iff A is a crisp set F2: F3: f(A) assumes the maximum value iff A is maximally fuzzy.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 60 Measurement Based on Distance nearest crisp set One measure of fuzziness is defined in terms of a metric distance from the set A to the nearest crisp set. Distance from point A(a 0,a 1,…,a n ) to B(b 0,b 1,…,b n )

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 61 Measurement Based on Distance If p=2 the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 62 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.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 63 Classic Power Set Classic Power Set: the set of all subsets of a classic set. Let consider X={x 1,x 2,x 3 } Power Set is represented by 2 |X| 2 |X| ={ , {x 1 }, {x 2 }, {x 3 }, {x 1,x 2 }, {x 1,x 3 }, {x 2,x 3 }, X}

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 64 Vertices The 8 sets can correspond to 8 vectors 2 |X| ={ , {x 1 }, {x 2 }, {x 3 }, {x 1,x 2 }, {x 1,x 3 }, {x 2,x 3 }, 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 65 The vertices in space

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 66 Fuzzy Power Set The Fuzzy Power set is the set of all fuzzy subsets of X={x 1,x 2,x 3 } 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 67 The Fuzzy Cube

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 68 Fuzzy Operations Let X={x 1,x 2 } and A={(x 1,1/3),(x 2,3/4)} Let A´ represent the complement of A A´={(x 1,2/3),(x 2,1/4)} A  A´={(x 1,2/3),(x 2,3/4)} A  A´={(x 1,1/3),(x 2,1/4)}

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 69 Fuzzy Operations in the Space

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 70 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 S  not-S and not-S  S t(S)=t(not-S)=1-t(S)  t(S)=0.5

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 71 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|

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 72 Distance The distance d p 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 73 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

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 74 Fuzzy Cardinality

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 75 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.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 76 Fuzzy Operations in the Space

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 77 Fuzzy Entropy Geometry

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 78 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= {(x 1,0.5),(x 2,0.5)} 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…

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 79 Subsets Sets contain subsets. A is a subset of B (A  B) 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 (A  B iff A  2 |B| ).

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 80 Subsethood examples Consider A={(x 1,1/3),(x 2 =1/2)} and B={(x 1,1/2),(x 2 =3/4)} A  B, but B  A

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 81 Not Fuzzy Subsethood The so called membership dominated definition is not fuzzy. The fuzzy power set of B (F(2 B )) is the hyper rectangle docked at the origin of the hyper cube. Any set is either a subset or not.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 82 Fuzzy power set size F(2 B ) has infinity cardinality. For finite dimensional sets the size of F(2 B ) is the Lebesgue measure or volume V(B)

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 83 Fuzzy Subsethood Let S(A,B)=Degree(A  B)=  F(2 B ) (A) Suppose only element j violates  A (x j )  B (x j ), so A is not totally subset of B. Counting violations and their magnitudes shows the degree of subsethood.

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 84 Fuzzy Subsethood Supersethood(A,B)=1-S(A,B) Sum all violations=max(0,  A (x j )-  B (x j )) 0  S(A,B)  1

@2005 Adriano Cruz NCE e IM - UFRJOperations of Fuzzy Sets 85 Subsethood measures Consider A={(x 1,0.5),(x 2 =0.5)} and B={(x 1,0.25),(x 2 =0.9)}