Summary. Operations on Fuzzy Sets. Zadeh s Definitions. Zadeh s Operations T-Norms S-Norms. Properties of Fuzzy Sets Fuzzy Measures

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1 Summary Operations on Fuzzy Sets Zadeh s Operations T-Norms S-Norms Adriano Cruz 00 NCE e IM/UFRJ Properties of Fuzzy Sets Fuzzy Measures One should not increase, beyond what is necessary, the number of entities required to explain anything. Occam's Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 3

2 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- Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 8 Intersection operation T-norms 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 µ A (x), µ B (x), µ C (x) and µ 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.1 T(0,0) = 0 T. 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 10 T-norms: comments Minimum, T-norm? 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 Τ(µ Α (x), µ Β (x)) <= min(µ Α (x), µ Β (x)) T.1 min(0,0) = 0 T. 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 <= Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 1

3 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 µ A (x), µ B (x), µ C (x) e µ 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. 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 14 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(µ Α (x), µ Β (x)) >= max(µ Α (x), µ Β (x)) S-norms: comments 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 16 Maximum, S-norm? Algebraic sum, S-norm? S.1 max(0,0) = 0 S. 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 µ A B a,b =a+b a b S S. a+b a b=b+a b a comutativa S. 3 a+ 0 a 0 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 18

4 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 Other examples Prove that T(a,b) <= min(a,b) T5: T(a,b) <= T(a,1) = a T: T(a,b) = T(b,a) T5: T(b,a) <= T(b,1) = b T: T(a,b) <= Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 0 Pairs of T-norms and S-norms T-norm - Drastic Product: min x,y if max x,y 1 DP x,y 0 x,y 1 S-norm - Drastic Sum: max x,y if min x,y 0 DS x,y 1 x,y> 0 Pairs of T-norms and S-norms T-norm - Bounded Difference: BD x,y max 0, x+y 1 S-norm - Bounded Sum: BS x,y min 1, Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Pairs of T-norms and S-norms T-norm Einstein Product: Pairs of T-norms and S-norms T-norm Algebraic Product: xy EP x,y x+y xy S-norm - Einstein Sum: ES x,y x+y 1+x. y AP x,y =xy S-norm - Algebraic Sum: AS x,y =x+y Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 4

5 Pairs of T-norms and S-norms Four T-norm operators T-norm Hamacher Product: HP x,y xy x+y xy S-norm - Hamacher Sum: x+y xy HS x,y 1 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 6 Four T-conorm operators Pairs of T-norms and S-norms T-norm Dubois-Prade: xy DPr T x,y max p,x,y Obs. p is a parameter that ranges from 0 to 1. S-norm Dubois-Prade: x+y xy min 1 p,x,y DPr S x,y max p,1 x,1 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 8 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) Pairs of T-norms and S-norms T-norm Yager: Y T x,y 1 min 1, 1 x p 1 y p 1 p Obs. p is a parameter that ranges from 0 to. S-norm Yager: Y T x,y min 1, x p +y p 1 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 30

6 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)) Complement When p-> Yager T-norm converges to min(x,y) Yager S-norm converges to Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 3 Fuzzy Complement axioms Fuzzy Complements A fuzzy complement operator is a continuous function N:[0,1] :[0,1] [0,1] which meets the following axioms: N(0)=1 and N(1) = 0 (boundary) (a.1) N(a) N(b) ) if a b (monotonicity) (a.) Another optional requirements are N(x) is continuous (a.3) N(N(a))= ))=a (involution) (a.4) Continuous a.3 All functions a.1 e a. Involutive a.4 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 34 Usual Fuzzy Complement Consider - N(x)=1-x N(0)=1 and N(1) = 0 (boundary) N(a) N(b) if a b N(N(x))=1-(1-x)= )=x Continuous in the interval (monotonicity) Another Ex of Complement 1 for a t Consider N x 0 a>t 1 Example of complement 0.9 t,a 0,1 N (x) Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 36

7 Another Ex of Complement 1 Consider N x 1 for a t 0 a>t t,a 0,1 Satisfy only the axiomatic requirements: N(1)=0, N(0)=1 N(x) ) is monotonic N(x) ) is not continuous N(x) ) is not involutive Sugeno s complement The operator is defined as N s a 1 a 1+sa where s is a parameter greater than 1. For each s, we obtain a particular Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 38 Sugeno s complement Yager s complement The operator is defined as N y a 1 a y 1 y where y is a positive Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 40 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 4

8 Fuzzy Sets Properties Properties Comutativity Associativity Distributivity Absorption A B=B A A B=B A A B C A B C A B C A B C A B C A B A C A B C A B A C A A B =A A A B Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 44 Fuzzy Sets Properties Identity De Morgan A =A A A X=X A X=A A B A B A B A Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 45 Checking Properties Remember that a if a b min a,b = b if b<a a+b a b min a,b = and b if a b max a,b = a if b<a a+b+ a b max a,b Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 46 Checking Properties Checking Properties Lets check the absorption property A A B =A max µ A x,min µ A x,µ B x =µ A x A B = A A B = a+b a b a+ a+b a b a a+b a b A A B = = If a b A A B = A A B = 3a+b a b a b+ a b 3a +b a b a b+ a b 4 3a +b a b a b a b 4 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 48

9 Checking Properties Laws of Aristotle A A B = if a<b A A B = A A B = 3a +b a b a b+ a b 4 3a +b+ a b a b a b 4 a 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 A Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 50 Law of non-contradiction Laws of Aristotle 1.0 Non adults adults Law of excluded middle: for any proposition P, it is true that (P or not-p). adults adults So the union of a set and its complement should give all the universe adults adults However the result may be not the universe A A Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 5 Law of non-contradiction 1.0 Non adults adults Measuring adults adults Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 54

10 Fuzzy Entropy Fuzzy Entropy The entropy of a fuzzy set is defined as c A A E A c A A 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 55 1 The entropy of the adult fuzzy set is c A A = 5 c A A = = 35 E A = 5 35 = 0.14 No adults adults adult Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 56 Fuzziness Measurements A measure of fuzziness is a function f : X 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 Requirements F1: f(a) = 0 iff A is a crisp set A B means A is less fuzzy than B F: if A B then f A f B F3: f(a) assumes the maximum value iff A is maximally Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 58 Measurement Based on Distance 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 ) d p A,B p n i=1 a i b i p Measurement Based on Distance If p= the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance. d p A,B p n i=1 a i b i Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 60

11 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 61 Classic Power Set Classic Power Set: the set of all subsets of a classic set. Let consider X={x 1,x,x 3 } Power Set is represented by X X ={, {x 1 }, {x }, {x 3 }, {x 1,x }, {x 1,x 3 }, {x,x 3 }, Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 6 Vertices The vertices in space The 8 sets can correspond to 8 vectors (0,0,1) x3 (1,0,1) X ={, {x 1 }, {x }, {x 3 }, {x 1,x }, {x 1,x 3 }, {x,x 3 }, X} (0,1,1) (1,1,1) 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 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 63 (0,1,0) x x (0,0,0) 1 (1,0,0) Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 64 Fuzzy Power Set The Fuzzy Cube The Fuzzy Power set is the set of all fuzzy (0,0,1) x 3 (1,0,1) subsets of X={x 1,x,x 3 } It is represented by F( X ) A Fuzzy subset of X is a point in a cube (0,1,1) 0.7 (1,1,1) A={(x 1,0.5),(x,0.3),(x 3,0.7)} The Fuzzy Power set is the unit hypercube 0.3 x (0,1,0) 0.5 (1,1,0) x1 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 66

12 Fuzzy Distances Let X={x 1,x } and A={(x 1,1/3),(x,3/4)} Maximal Fuzziness x Let X={x 1 1,x } and A={(x 1,1/),(x,1/)} x 1 n=1 3/4 n= A={(x1,1/3), (x,3/4)} /4 A={(x1,1/), (x,1/)} /4 n= n=1 1/4 0,0 1/ 1 x1 0,0 1/3 /3 1 Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 68 Fuzzy Operations Fuzzy Operations in the Space Let X={x 1,x } and A={(x 1,1/3),(x,3/4)} Let A represent the complement of A A ={(x 1,/3),(x,1/4)} (0,1) 3/4 x A(1/3,3/4) A A (1,1) A A ={(x 1,/3),(x,3/4)} A A ={(x 1,1/3),(x,1/4)} 1/4 φ A A 1/3 /3 A (/3,1/4) (1,0) x Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations 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 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 n A µ A x i Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 7

13 Distance The distance d p between two sets represented by points in the space is defined as d p A,B p n i=1 µ A x i µ B x i p If p= the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance Distance and Cardinality If the point B is the empty set (the origin) d 1 A,O µ A x i 0 i=1 So the cardinality of a fuzzy set is the Hamming distance to the origin n d 1 A,O A µ A x i n i= Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 74 Fuzzy Cardinality Fuzzy Entropy (0,1) x (1,1) How fuzzy is a fuzzy set? 3/4 A Fuzzy entropy varies from 0 to 1. Cube vertices has entropy 0. A =d 1 (A,φ) The middle point has entropy 1. φ 1/3 (1,0) x Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 76 Fuzzy Operations in the Space Fuzzy Entropy Geometry (0,1) 3/4 x A A A (1,1) (0,1) 3/4 x a A (1,1) 1/4 A A A b φ (1,0) 1/3 /3 x1 A A E A A Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 77 φ 1/3 E A a b d1 A,A near d 1 A,A far Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 78 x1

14 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,0.5)} E(A)=0.5/0.5=1 Let T(x,y)=x.y and C(x,y)=x+y-xy E(A)=0.5/0.75=0.333 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 B Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 80 Subsethood examples Consider A={(x 1,1/3),(x =1/)} and B={(x 1,1/),(x =3/4)} A B, but B A (0,1) 3/4 1/ x A B (1,1) Not Fuzzy Subsethood The so called membership dominated definition is not fuzzy. The fuzzy power set of B (F( B )) is the hyper rectangle docked at the origin of the hyper cube. Any set is either a subset or not. φ 1/3 1/ (1,0) Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 8 Fuzzy power set size F( B ) has infinity cardinality. For finite dimensional sets the size of F( B ) is the Lebesgue measure or x volume V(B) (1,1) (0,1) n V B µ B x i i=1 3/4 1/ A B Fuzzy Subsethood Let S(A,B)=Degree(A B)=µ F( B ) (A) Suppose only element j violates µ A (x j ) µ B (x j ), so A is not totally subset of B. Counting violations and their magnitudes shows the degree of subsethood. (1,0) φ 1/3 1/ x Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 84

15 Fuzzy Subsethood Supersethood(A,B)=1-S(A,B) Sum all violations=max(0,µ A (x j )-µ B (x j )) 0 S(A,B) 1 x X max 0, µ A x µ B x Supersethood A,B A max 0, µ A x µ B x S A,B 1 x X A Subsethood measures Consider A={(x 1,0.5),(x =0.5)} and B={(x 1,0.5),(x =0.9)} max 0, max 0, S A,B = S B,A = 1 S A,B max 0, max 0, S B,A = Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM - UFRJ Operations of Fuzzy Sets 86

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