Lecture 4 (Fuzzy Set Operations)
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1 Lecture 4 (Fuzzy Set Operation) We need a radically different kind of mathematic, the mathematic of fuzzy or cloudy quantitie which are not decribable in term of probability ditribution L.A.Zadeh, 1962 صفحه ۱
2 Some point of the previou lecture Fuzzy Logic i ome kind of multi-valued logic. Unlike crip two valued logic, the truth value in fuzzy logic can be any number between 0 and 1 and hence i an extenion to claical logic Conequently, many of the Natural Language propoition can be repreented with fuzzy logic صفحه ۲
3 A fuzzy et (A generalized concept of the conventional crip et) i pecified with a memberhip function A (x) which repreent degree of memberhip in the et. A ={ (x, A (x)) xu x U, 00 A (x) 1} صفحه ۳
4 Baic Set-Theoretic Operation Equality: Subet: Complement: Union: Interection: A B A B A B A X A ( x ) 1 ( x ) C A B ( x ) max( ( x ), ( x )) ( x ) ( x ) c A B A B C A B ( x ) min( ( x ), ( x )) ( x ) ( x ) A c A B A B A B A Slide for fuzzy et,, J.-. Roger Jang صفحه ۴
5 De Morgan Law Set operation on fuzzy et are uppoed to be defined uch that the more previou known law and equalitie remain true in fuzzy et a well. Having the previou definition for fuzzy et operation, we can verify De Morgan law in fuzzy logic a well. A B A B A B A B Exercie: Prove the above equalitie صفحه ۵
6 Fuzzy Set operation in detail - Fuzzy Complement Fuzzy complement i actually a function ay c that map the memberhip function A (x) to the memberhip function of A (x) the complement et. A ( x) c c[ ( x)] ( x) Definition: Any function c :[0,1][0,1] that atifie the following Axiom c1 and c2 i called a fuzzy complement A A صفحه ۶
7 Requirement Axiom c1. (boundary condition) c( 0) 1, c(1) Axiom c2. (non-increaing condition) 0 a, b [0,1] if a b c( a) c( Axiom c1 require that if an element belong to a fuzzy et to degree zero (one), then it hould belong to the complement of thi fuzzy et to degree one (zero). Axiom c2 mean that an increae in memberhip value of a fuzzy et mut reult in a decreae or no change in memberhip value of the complement et صفحه ۷
8 Clearly, in claical crip logic (where domain of definition of the complement function i {0,1}) there i only one complement function which atifie the above axiom wherea in fuzzy logic, there are many function with a domain [0,1] which atify the above condition. Example of fuzzy complement 1. Baic Fuzzy Complement c[ ( x)] 1 ( x) or c[ a] 1a A A صفحه ۸
9 2. Sugeno cla of fuzzy complement c 1 a ( a ) ( 1, 1 a ) For any value of the parameter, a particular fuzzy complement function i obtained 3. Yager cla of fuzzy complement 1 c ( a) (1 a ) (0, ) For any value of the parameter, a particular fuzzy complement function i obtained صفحه ۹
10 Graphical Repreentation of the Sugeno Cla Complement 1 Sugeno Cla of Fuzzy Complement for different value of Landa landa=0 landa=-0.5 landa= landa=2 0.2 landa=8 0.1 landa= صفحه ۱۰
11 Graphical Repreentation of the Yager Cla Complement 1 Yager Cla of Fuzzy Complement for different value of w w=7 0.6 w= w=1 0.3 w= w= w= صفحه ۱۱
12 - Fuzzy Union -norm (t-conorm) Intuitively, the union of two et, AB A B mean a fuzzy et (in particular the mallet one) containing both A and B. The union of two fuzzy et can be defined with a function named -norm :[0,1]x[0,1][0,1] which map the memberhip function of fuzzy et A and B into the memberhip function of the union of A and B [ ( x), ( x)] ( x) A B AB (called AB) The requirement for a function to be an -norm are a follow: صفحه ۱۲
13 Axiom 1. (boundary condition) (1,1) 1, (0, a ) ( a,0 ) a Axiom 2. (commutative condition) ( a, b ) ( b, a Axiom 3. (non-decreaing condition) ) if a a and b b ( a, b ) ( a, b ) Axiom 4. (aociative condition) ( ( a, b ), c ) ( a, ( b, c )) صفحه ۱۳
14 Definition: Any function :[0,1]x[0,1][0,1] that atifie the above 4 axiom i called an -norm Example of fuzzy -norm 1. Dombi cal ( 0, ) 1 1[( a 1) 1 1 ( 1) b ] 1 2. Duboi-Prade cal (0,1) a b ab min b,1 ) max( 1 a,1 b, ) صفحه ۱۴
15 3. Yager cal ( 0, ) min[ 1,( a b ) 1 ] 4. Dratic Sum: 5. Eintein Sum: a d ( a, b ) b 1 e a b 1 ab if if b a otherwie 0 0 صفحه ۱۵
16 6. Algebraic Sum: a b ab a 7. Maximum (Baic fuzzy Union) max max Theorem S1: For any -norm, (a, the following inequality hold: (for any a,b [0,1] max( a, b ) ( a, ) ( a It mean that the mallet -norm (or mallet union of two fuzzy et) i maximum while the larget -norm i Dratic um b d, b ) صفحه ۱۶
17 Theorem S2: Dombi -norm and Yager -norm cover the whole pectrum of -norm when their parameter change In it extreme cae: im And Alo im max(a, max(a, im 0 im So it i poible to build any -norm with chooing the right parameter in any of the yager or dombi -norm 0 d d صفحه ۱۷
18 - Fuzzy Interection t-norm Intuitively, the interection of two et, AB A mean a fuzzy et (in particular the larget one) containing by both A and B. The Interection of two fuzzy et can be defined with a function named t-norm t t:[0,1]x[0,1][0,1] which map the memberhip function of fuzzy et A and B into the memberhip function of the interection of t[ ( x), ( x)] ( x) A and B A B AB The requirement for a function to be a t-norm are a follow: صفحه ۱۸
19 Axiom t1. (boundary condition) t (0,0 ) 0, t ( a,1) t (1, a ) a Axiom t2. (commutative condition) t ( a, b ) t ( b, a Axiom t3. (non-decreaing condition) ) if a a and b b t ( a, b ) t ( a, b ) Axiom t4. (aociative condition) t ( t ( a, b ), c ) t ( a, t ( b, c )) صفحه ۱۹
20 Definition: Any function t:[0,1]x[0,1][0,1] that atifie the above 4 axiom i called a t-normt Example of fuzzy t-norm 1. Dombi cal ( 0, ) t 1 1[( a 1) 1 1 ( 1) b ] 1 2. Duboi-Prade cal (0,1) t ab max b, ) صفحه ۲۰
21 3. Yager cal ( 0, ) t 1 min[ 1,((1 a) (1 ) 1 ] 4. Dratic Product: a t dp ( a, b ) b 0 if b if a otherwie Eintein Product: t ep 2 ( a ab b a صفحه ۲۱
22 6. Algebraic Product: t ab ap 7. Minimum (Baic fuzzy Interection) min min Theorem T1: For any t-norm, t(a, the following inequality hold: (for any a,b [0,1] ) t dp t min b ) It mean that the larget t-norm (or larget interection of two fuzzy et) i minimum while the mallet t-norm i Dratic product صفحه ۲۲
23 Theorem T2: Dombi t-norm t and Yager t-norm t cover the whole pectrum of t-norm t when their parameter change In it extreme cae: imt min(a, im t 0 t dp And alo: imt min(a, im t 0 dp So it i poible to build any t-norm with chooing the right parameter in any of the yager or dombi t-norm. صفحه ۲۳
24 Graphical repreentation of theorem S1 and S2 max ( a, d AB (x) Algebraic um: S(a,= a (a, A (x) B (x) Yager: S(a,= w (a, W=3 S(a,=max(a, صفحه ۲۴
25 Graphical repreentation of theorem T1 and T2 t dp t( a, min AB (x) S(a,=min(a, A (x) B (x) Yager: t(a,=t w (a, W=3 Algebraic product: S(a,= ap (a, صفحه ۲۵
26 Generalized De Morgan Law Uing the new definition of -norm and t-norm t intead of the baic fuzzy union and baic fuzzy interection repectively, the generalized De Morgan Law can be hown a follow: c( t(a, ) = ( c(a), c( ) c( (a, ) = t( c(a), c( ) where c(.) denote for any fuzzy complement and (.) and t(.) denote for fuzzy -norm and fuzzy t-norm t repectively. صفحه ۲۶
27 Aociated cla An -norm (a,, a t-norm t t(a, and a fuzzy complement c(a) form an aociated cla if they all together atify the Generalized De Morgan law c[(a,]=t[c(a),c(] It can be hown that there i a t-norm t aociated with each -norm in the ene that there i a complement uch that the De Morgan law are atified. For example the Yager -norm and t-norm t are aociated with each other through baic fuzzy complement صفحه ۲۷
28 Reference 1. L.X. Wang, A coure in Fuzzy Sytem and control 2. Tutorial on Fuzzy Logic,, Jan Jantzen, Technical Univerity of Denmark, Technical report no 98-E E 868, Slide for fuzzy et,, J.-. Roger Jang صفحه ۲۸
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