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FUZZY SETS AND FUZZY LOGIC Theory and Applications PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations Fuzzy complements • Axiomatic skeleton Axiom c1. c0 1 and c1 0 (boundary conditions). Axiom c2. For all a, b [0,1] if a b , then c(a) c(b) (monotonicity). Fuzzy complements • Desirable requirements Axiom c3. c is a continuous function. Axiom c4. c is involutive, which means that c(c(a)) a for each a [0,1]. Fuzzy complements • Theorem 3.1 Let a function c : [0, 1] [0, 1] satisfy Axioms c2 and c4. Then, c also satisfies Axioms c1 and c3. Moreover, c must be a bijective function. Fuzzy complements Fuzzy complements Fuzzy complements • Sugeno class 1 a c (a ) , where (1, ). 1 a • Yager class cw (a) (1 a ) , where w (0, ). w 1/ w Fuzzy complements Fuzzy complements • Theorem 3.2 Every fuzzy complement has at most one equilibrium. Fuzzy complements • Theorem 3.3 Assume that a given fuzzy complement c has an equilibrium ec , which by Theorem 3.2 is unique. Then a ca iff a ec and a ca iff a ec . Fuzzy complements • Theorem 3.4 If c is a continuous fuzzy complement, then c has a unique equilibrium. • Theorem 3.5 If a complement c has an equilibrium ec , then d ec ec . Fuzzy complements Fuzzy complements • Theorem 3.6 For each a [0, 1] , a c(a) iff c(c(a)) a , that is, when the complement is involutive. d Fuzzy complements • Theorem 3.7 (First Characterization Theorem of Fuzzy Complements). Let c be a function from [0, 1] to [0, 1]. Then, c is a fuzzy complement (involutive) iff there exists a continuous function g from [0, 1] to R such that g (0) 0 , g is strictly increasing, and c(a) g 1 ( g (1) g (a)) for all a [0, 1]. Fuzzy complements • Increasing generators Sugeno: g a 1 ln1 a for 1. Yager: gw a a w for w 0. Fuzzy complements • Theorem 3.8 (Second Characterization Theorem of Fuzzy complements). Let c be a function from [0, 1] to [0, 1]. Then c is a fuzzy complement iff there exists a continuous function f from [0, 1] to R such that f 1 0 , f is strictly decreasing, and ca f 1 f 0 f a for all a [0, 1] . Fuzzy complements • Decreasing generators Sugeno: f (a) ln(1 a) where 1. 1 ln(1 a), Yager: f (a) 1 a w , where w 0. Fuzzy intersections: t-norms • Axiomatic skeleton Axiom i1. ia,1 a (boundary condition). Axiom i2. b d implies ia, b ia, d (monotonicity). Fuzzy intersections: t-norms • Axiomatic skeleton Axiom i3. ia, b ib, a (commutativity). Axiom i4. ia, ib, d iia, b, d (associativity). Fuzzy intersections: t-norms • Desirable requirements Axiom i5 i is a continuous function (continuity). Axiom i6 ia, a a (subidempotency). Axiom i7 a1 a2 and b1 b2 implies i(a1 , b1 ) i(a2 , b2 ) (strict monotonicity). Fuzzy intersections: t-norms • Archimedean t-norm: A t-norm satisfies Axiom i5 and i6. • Strict Archimedean t-norm: Archimedean t-norm and satisfies Axiom i7. Fuzzy intersections: t-norms • Frequently used t-norms Standard intersection : i (a, b) min(a, b). Algebraic product : i (a, b) ab. Bounded difference: ia, b max(0, a b 1) a when b 1 Drasticintersection : i (a, b) b when a 1 0 otherwise. Fuzzy intersections: t-norms Fuzzy intersections: t-norms Fuzzy intersections: t-norms • Theorem 3.9 The standard fuzzy intersection is the only idempotent t-norm. • Theorem 3.10 For all a, b [0, 1] , imin a, b ia, b mina, b, where imin denotes the drastic intersection. Fuzzy intersections: t-norms • Pseudo-inverse of decreasing generator The pseudo-inverse of a decreasing (1) generator f , denoted by f , is a function from R to [0, 1] given by for a (, 0) 1 1 ( 1) f (a) f (a) for a [0, f (0)] 0 for a ( f (0), ) (1) f where is the ordinary inverse of f . Fuzzy intersections: t-norms • Pseudo-inverse of increasing generator The pseudo-inverse of a increasing (1) g generator , denoted by g , is a function from R to [0, 1] given by for a (, 0) 0 1 ( 1) g (a ) g (a ) for a [0, g (1)] 1 for a ( g (1), ) (1) g where is the ordinary inverse of g. Fuzzy intersections: t-norms • Lemma 3.1 Let f be a decreasing generator. Then a function g defined by g (a) f (0) f (a) for any a [0, 1] is an increasing generator with g (1) f (0), and its pseudo-inverse g (1) is given by g ( 1) (a) f ( 1) ( f (0) a) for any a R. Fuzzy intersections: t-norms • Lemma 3.2 Let g be a increasing generator. Then a function f defined by f (a) g (1) g (a) for any a [0, 1] is an decreasing generator with f (0) g (1), and its pseudo-inverse f (1) is given by f ( 1) (a) g ( 1) ( g (1) a) for any a R. Fuzzy intersections: t-norms • Theorem 3.11 (Characterization Theorem of t-Norms). Let i be a binary operation on the unit interval. Then, i is an Archimedean t-norm iff there exists a decreasing generator f such that i(a, b) f for all a, b [0, 1]. ( 1) ( f (a) f (b)) Fuzzy intersections: t-norms • [Schweizer and Sklar, 1963] f p (a) 1 a p ( p 0). where z (, 0) 1 ( 1) f p ( z ) (1 z )1 p where z [0, 1] 0 where z (1, ) i p (a, b) f p( 1) ( f p (a) f p (b)) f p( 1) (2 a p b p ) (a p b p 1)1 p when 2 a p b p [0, 1] otherwise. 0 (max(0, a p b P 1))1 p . Fuzzy intersections: t-norms • [Yager, 1980f] f w (a) (1 a) w ( w 0), 1w where z [0, 1] 1 z ( 1) f w ( z) where z (1, ) 0 iw (a, b) f w( 1) ( f w (a ) f w (b)) f w( 1) ((1 a) w (1 b) w ) 1 ((1 a) (1 b) ) 0 w w 1w when (1 a ) w (1 b) w [0, 1] otherwise. 1 min(1, [(1 a) w (1 b) w ]1 w ). Fuzzy intersections: t-norms • [Frank, 1979] sa 1 f s (a) ln ( s 0, s 1), s 1 ( 1) f s ( z ) logs (1 ( s 1)e z ). is (a, b) f s( 1) ( f s (a) f s (b)) f ( 1) s ( s a 1)(s b 1) ln 2 ( s 1 ) ( s a 1)(s b 1) logs 1 ( s 1) 2 ( s 1 ) ( s a 1)(s b 1) logs 1 . s 1 Fuzzy intersections: t-norms • Theorem 3.12 Let iw denote the class of Yager t-norms. Then, imin (a, b) iw (a, b) min(a, b) for all a, b [0, 1], where the lower and upper bounds are obtained for w 0 and w ,respectively. Fuzzy intersections: t-norms • Theorem 3.13 Let i be a t-norm and g : [0, 1] [0, 1] be a function such that g is strictly increasing and continuous in (0, 1) and g (0) 0, g (1) 1. Then, the function i g defined by i (a, b) g g ( 1) (i( g (a), g (b))) for all a, b [0, 1] ,where g (1) denotes the pseudo-inverse of g , is also a t-norm. Fuzzy unions: t-conorms • Axiomatic skeleton Axiom u1. u(a, 0) a (boundary condition). Axiom u2. b d impliesu(a, b) u(a, d ) (monotonici ty). Fuzzy unions: t-conorms • Axiomatic skeleton Axiom u3. u(a, b) u(b, a) (commutativ ity). Axiom u4. u(a, u(b, d )) u(u(a, b), d ) (associativity). Fuzzy unions: t-conorms • Desirable requirements Axiom u5. u is a continuousfunction(continuity). Axiom u6. u(a, a) a (superidempotency). Axiom u7. a1 a2 and b1 b2 implies u(a1, b1 ) u(a2 , b2 ) (strict monotonicity). Fuzzy unions: t-conorms • Frequently used t-conorms for all a, b [0, 1] Standard union : u (a, b) max(a, b). Algebraic sum : u (a, b) a b ab. Bounded sum : u (a, b) min(1, a b). a when b 0 Drasticunion : u (a, b) b when a 0 1 otherwise. Fuzzy unions: t-conorms Fuzzy unions: t-conorms Fuzzy unions: t-conorms • Theorem 3.14 The standard fuzzy union is the only idempotent t-conorm. Fuzzy unions: t-conorms • Theorem 3.15 For all a, b [0, 1], max(a, b) u(a, b) umax (a, b). Fuzzy unions: t-conorms • Theorem 3.16 (Characterization Theorem of t-Conorms). Let u be a binary operation on the unit interval. Then, u is an Archimedean tconorm iff there exists an increasing generator such that u(a, b) g for all a, b [0, 1]. ( 1) ( g (a) g (b)) Fuzzy unions: t-conorms • [Schweizer and Sklar, 1963] g p (a ) 1 (1 a ) p ( p 0). 1 p when z [0, 1] 1 ( 1 z ) ( 1) g p ( z) when z (1, ) 1 u p (a, b) g (p1) (1 (1 a ) p 1 (1 b) p ) 1 [(1 a ) p (1 b) p 1]1 p when 2 (1 a ) p (1 b) p [0, 1] otherwise. 1 1 max(0, (1 a) p (1 b) p 1) 1 p . Fuzzy unions: t-conorms • [Yager, 1980f] g w (a ) a w ( w 0), 1w when z [0, 1] z ( 1) g w ( z) 1 when z (1, ) u w (a, b) g w( 1) (a w b w ) min(1, (a w b w )1 w ). Fuzzy unions: t-conorms • [Frank, 1979] s1 a 1 g s (a ) ln ( s 0, s 1) s 1 g s( 1) ( z ) 1 logs (1 ( s 1)e z ), ( s1 a 1)(s1b 1) u s (a, b) 1 logs 1 . s 1 Fuzzy unions: t-conorms • Theorem 3.17 Let uw denote the class of Yager t-conorms. max(a, b) uw (a, b) umax (a, b) for all a, b [0, 1] where the lower and upper bounds are obtained for w and w 0 , respectively. Fuzzy unions: t-conorms • Theorem 3.18 Let u be a t-conorm and let g : [0, 1] [0, 1] be a function such that g is strictly increaning and continuous in (0, 1) and g (0) 0, g (1) 1. g u Then, the function defined by u (a, b) g g ( 1) (u( g (a), g (b))) for all a, b [0, 1] is also a t-conorm. Combinations of operators • Theorem 3.19 The triples 〈min, max, c〉and〈imin, umax, c〉are dual with respect to any fuzzy complement c. Combinations of operators • Theorem 3.20 Given a t-norm i and an involutive fuzzy complement c, the binary operation u on [0, 1] defined by u(a, b) c(i(c(a), c(b))) for all a, b [0, 1] is a t-conorm such that 〈i, u, c〉is a dual triple. Combinations of operators • Theorem 3.21 Given a t-conorm u and an involutive fuzzy complement c, the binary operation i on [0, 1] defined by i(a, b) c(u(c(a), c(b))) for all a, b [0, 1] is a t-norm such that 〈i, u, c〉is a dual triple. Combinations of operators • Theorem 3.22 Given an involutive fuzzy complement c and an increasing generator g of c, the t-norm and t-conorm generated by g are dual with respect to c. Combinations of operators • Theorem 3.23 Let〈i, u, c〉be a dual triple generated by Theorem 3.22. Then, the fuzzy operations i, u, c satisfy the law of excluded middle and the law of contradiction. Combinations of operators • Theorem 3.24 Let〈i, u, c〉be a dual triple that satisfies the law of excluded middle and the law of contradiction. Then,〈i, u, c〉does not satisfy the distributive laws. Aggregation operations • Axiomatic requirements Axiom h1. h(0, 0, ..., 0) 0 and h(1, 1, ...,1) 1 (boundary onditions c ). Axiom h2. For any pair a1 , a2 , ..., an and b1 , b2 , ..., bn of n - tuples such thatai , bi [0 , 1] for all i N n , if ai bi for all i N n , then h(a1, a2 , ..., an ) h(b1, b2 , ..., bn ) ; thatis, h is m onotonicincreasing in all its arguments. Aggregation operations • Axiomatic requirements Axiom h3. h is continuous function. Aggregation operations • Additional requirements Axiom h4. h is a sym m etric functionin all its arguments;thatis, h(a1, a2 , ..., an ) h(a p (1) , ap ( 2) , ..., ap ( n ) ) for any permutation p on N n . Axiom h5. h is an idem potentfunction;thatis, h(a, a, ...,a) a for all a [0, 1]. Aggregation operations • Theorem 3.25 Let h : [0,1]n R be a function hat t satisfiesAxiomh1,Axiom h2, and theproperty h(a1 b1, a2 b2 , ..., an bn ) h(a1, a2 , ..., an ) h(b1, b2 , ..., bn ) where ai , bi , ai bi [0,1] for all i N n . T hen, n h(a1, a2 , ..., an ) wi ai , i 1 where wi 0 for all i N n. Aggregation operations • Theorem 3.26 Let h : [0,1]n [0,1] be a function hat t satisfiesAxiomh1,Axiom h3, and theproperty h(max(a1 , b1 ) , ..., max(an , bn )) max(h(a1, a2 , ..., an ), h(b1, b2 , ..., bn )) hi ( hi (ai )) hi (ai ) where hi (ai ) h(0,...,0,ai , 0, ..., 0) for all i N n . T hen, h(a1, a2 , ..., an ) max(min(w1 , a1 ), ..., min(wn , an )), where wi [0,1] for all i N n . Aggregation operations • Theorem 3.27 Let h : [0,1]n [0,1] be a function hat t satisfiesAxiomh1,Axiom h3, and theproperty h(min(a1 , b1 ) , ..., min(an , bn )) min(h(a1, a2 , ..., an ), h(b1, b2 , ..., bn )) hi (ab) hi (a)hi (b) and hi (0) 0 where hi (ai ) h(1,...,1,ai , 1, ...,1) for all i N n . T hen,thereexist numbers α1, α2 , ...,αn [0,1] such that α α α h(a1, a2 , ..., an ) min(a1 1 , a2 2 , ..., an n ). Aggregation operations • Theorem 3.28 Let a normoperation h be continuousand idempotent. T hen,thereexists [0,1] such that max(a, b) where a, b [0, ] h(a, b) min(a, b) where a, b [ , 1] otherwise for any a, b [0,1]. Exercise 3 • • • • 3.6 3.7 3.13 3.14