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Chapter 2 The Operations of Fuzzy Set Outline • • • • • Standard operations of fuzzy set Fuzzy complement Fuzzy union Fuzzy intersection Other operations in fuzzy set Disjunctive sum Difference Distance Cartesian product • T-norms and t-conorms Standard operation of fuzzy set • Complement A ( x) 1 A ( x), x X 3 Standard operation of fuzzy set • Union AB ( x) max( A ( x), B ( x)), x X Standard operation of fuzzy set • Intersection AB ( x) min( A ( x), B ( x)), x X Fuzzy complement • C:[0,1][0,1] Fuzzy complement Fuzzy complement • Axioms C1 and C2 called “axiomatic skeleton ” are fundamental requisites to be a complement function, i.e., for any function C:[0,1][0,1] that satisfies axioms C1 and C2 is called a fuzzy complement. • Additional requirements Fuzzy complement • Example 1 : Standard function Axiom C1 Axiom C2 Axiom C3 Axiom C4 Fuzzy complement • Example 2 : Axiom C1 Axiom C2 X Axiom C3 X Axiom C4 Fuzzy complement • Example 3: Axiom C1 Axiom C2 Axiom C3 X Axiom C4 Fuzzy complement • Example 4: Yager’s function Axiom C1 Axiom C2 Axiom C3 Axiom C4 Fuzzy complement • Fuzzy partition If m subsets are defined in X, m-tuple (A1, A2,…,Am) holding the following conditions is called a fuzzy partition. Fuzzy union Fuzzy union • Axioms U1 ,U2,U3 and U4 called “axiomatic skeleton ” are fundamental requisites to be a union function, i.e., for any function U:[0,1]X[0,1][0,1] that satisfies axioms U1,U2,U3 and U4 is called a fuzzy union. • Additional requirements Fuzzy union • Example 1 : Standard function Axiom U1 Axiom U2 Axiom U3 Axiom U4 Axiom U5 Axiom U6 Fuzzy union • Example 2: Yager’s function Axiom U1 Axiom U2 Axiom U3 Axiom U4 Axiom U5 X Axiom U6 Fuzzy union Fuzzy union • Some frequently used fuzzy unions – Probabilistic sum (Algebraic Sum): U as ( x, y) x y x y – Bounded Sum (Bold union): Ubs ( x, y) min{1, x y} – Drastic Sum: max{x, y}, if min{x, y} 0 U ds ( x, y) 1, x, y 0 – Hamacher’s Sum x y (2 ) x y U hs ( x, y) , 0 1 (1 ) x y Fuzzy union Fuzzy intersection Fuzzy intersection • Axioms I1 ,I2,I3 and I4 called “axiomatic skeleton ” are fundamental requisites to be a intersection function, i.e., for any function I:[0,1]X[0,1][0,1] that satisfies axioms I1,I2,I3 and I4 is called a fuzzy intersection. • Additional requirements Fuzzy intersection • Example 1 : Standard function Axiom I1 Axiom I2 Axiom I3 Axiom I4 Axiom I5 Axiom I6 Fuzzy intersection • Example 2: Yager’s function Axiom I1 Axiom I2 Axiom I3 Axiom I4 Axiom I5 X Axiom I6 Fuzzy intersection Fuzzy intersection • Some frequently used fuzzy intersections – Probabilistic product (Algebraic product): I ap ( x, y) x y – Bounded product (Bold intersection): I bd ( x, y) max{0, x y 1} – Drastic product : min{x, y}, if max{x, y} 1 I dp ( x, y) 0, x, y 1 – Hamacher’s product x y I hp ( x, y ) , 0 (1 )(x y x y ) Fuzzy intersection Other operations • Disjunctive sum (exclusive OR) Other operations Other operations Other operations • Disjoint sum (elimination of common area) Other operations • Difference Crisp set Fuzzy set : Simple difference By using standard complement and intersection operations. Fuzzy set : Bounded difference Other operations • Example Simple difference Other operations • Example Bounded difference Other operations • Distance and difference Other operations • Distance Hamming distance Relative Hamming distance Other operations Euclidean distance Relative Euclidean distance Minkowski distance (w=1-> Hamming and w=2-> Euclidean) Other operations • Cartesian product Power Cartesian product Other operations • Example: – A = { (x1, 0.2), (x2, 0.5), (x3, 1) } – B = { (y1, 0.3), (y2, 0.9) } t-norms and t-conorms (s-norms) t-norms and t-conorms (s-norms) t-norms and t-conorms (s-norms) • Duality of t-norms and t-conorms Applying complements ( x, y) 1 T (1 x,1 y) 1 T ( x, y) T ( x, y), : t - conorms T : t - norms DeMorgan’s law