CHAPTER 2 OPERATIONS ON FUZZY SETS. Remark: 1. All functions that satisfy axioms c1 and c2 form the most general class of fuzzy complements.

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1 CHPTER OPERTIONS ON FUZZY SETS Fuzzy Complement Fuzzy Union Fuzzy Intersection Combinations of Operations General ggregation Operations Fuzzy Complement Remark: ll functions that satisfy axioms c and c form the most general class of fuzzy complements xioms c and c are called the axiomatic skeleton for fuzzy complements 3 xiom c is the ordinary complement for crisp sets Example: The threshold-type complement c ( a 0 for a t, for a > t, here a [ 0,] and t [0, ; t is called the threshold of c Def: complement of a fuzzy set is specified by a function c :[0,] [0,], hich assigns a value c( (x to each membership grade (x xiom: The function of a fuzzy complement must satisfy at least the folloing to requirements: c: c ( 0 and c ( 0 (boundary conditions c: For all a, b [0,], if a < b, then c( a c( (c is monotonic nonincreasing xiom: In practical significance, it is desirable to consider various additional requirements such as: c3 c is a continuous function - c4 c involutive, ie c ( c a a ( for all a [0,]

2 Example: n example that satisfies axioms c to c3 c( a ( + cosπa The yager class: ( a c ( a, here (, 0 Examples of involutive fuzzy complement: The sugeno class: a cλ ( a, + λa here λ (, Note: For λ 0, c ( a a λ For, c( a a -

3 Fuzzy Union Def: xioms: The union of to fuzzy sets and is specified by a function of the form: : [0,] [0,] [0,] Formally,, [ ] The function of a fuzzy union must satisfy at least the folloing axioms (the axiomatic skeleton for fuzzy set unions: Example: The Yager class union satisfies u to u5 [ ( a b ] u ( min, + here ( 0, Property:, u ( min(, a + b, u ( min(, a + b, u ( max( u u ( 0,0 0 ; u ( 0, u(,0 u(, (boundary conditions u u ( u( a (commutative u3 If a a and b b, then u ( u( a, b (monotonic u4 ( u(, c u( u( c u (associative xiom: It is often desirable to restrict the class of fuzzy union by considering various additional such as: u5 u is a continuous function u6 u ( a a, ie u is idempotent -3

4 Fuzzy Intersection Def: xioms: The intersection of to fuzzy sets and is specific by a function: Formally, i : [0,] [0,] [0,] [, ( ] x The function of fuzzy intersection must satisfy the folloing axioms (the axiomatic skeleton for fuzzy set intersections: Example: The Yager class union satisfies i to i4: here ( 0, Property: [ (( a + ( ] i ( min,, i ( min(, a b, i ( min(, ( a + (, i ( min( i i (, ; i ( 0, i(,0 i(0,0 0; (boundary conditions i i ( i( a ; (commutative i3 If a a and b b, then i ( i( a, b (monotonic i4 ( i(, c i( i( c i (associative The most important additional requirements: i5 i is a continuous function i6 i ( a a, ie i is idempotent -4

5 COMINTION OF OPERTORS: Theorem: max( u( umax (, When b 0, then u ( a umax ( a 0, then u ( b umax ( here proof: u max a ( b hen b 0, hen a 0, otherise 3 a, b (0,], then u ( u( u(, umax ( From,, 3, u ( u( max Using associativity (u4 ( u(0,0 u( u( 0,0 u y applying the boundary conditions (u ( u(,0,0 u ( 0 u a so u ( 0 a y monotonicity of u (u3 u ( u( 0 a y employing commutativity (u u ( u( a u( 0 b Hence u( max( Theorem: here i i min min ( i( min( a ( b 0 hen b, hen a, otherise proof: Similar to the previous theorem asing on the associativity (u4 ( i(, i( i(, i Using the boundary condition (u ( i(,, i ( i a so i ( a y monotonicity (u3 i ( i( a -5

6 With the commutativity (u i ( i( a i( b Left as an exercise! Property: The max operation is the loer bound of function u, ie the largest union The min operation is the upper bound of function i, ie the eakest intersection 3 Of all possible pairs of fuzzy sets unions and intersections, the max and min functions are closest to each other (an extreme pair, ie max( min( a b u( i( Proposition: The operations u max and u min is another extreme pair in the sense that u i max ( min ( u( i( The DeMorgan las are also satisfied under the complement operation c( a a u i max min ( imin( ( umax ( Proof: n exercise! The full scope of fuzzy aggregation operation: Proposition: The max and min functions together ith the complement satisfy the DeMorgan's las, ie max( min( min( max( Proof: See blackboard! c( a a Dombi Dombi 0 λ λ 0 Scheizer/Sklar Scheizer/Sklar - p p - Yager Yager 0 0 Generalized means - α i min i min max u u max Intersection verage Union -6

7 Theorem: u ( max( and i ( min( are the only continuous and idempotent fuzzy set union and fuzzy set intersection Proof: a The fuzzy set union: y associativity ( u( u( u( a u, The idempotency leads to(u6 ( u( u( u a, b [0,] u( α < u( α It contradicts to (** the assumption is not arrant ssume u ( a min(, it violates the boundary condition (u hen a 0 and b 3 ssume u ( b max(, it satisfies u and (* becomes u ( u( is also satisfied for all a < b (Remark: The Yager class satisfies continuity The intersection: n exercise! Similarly, Hence ( u(, u( u( u( u ( a u( u( u( b u,, (* When a (* is trival Wlog, Let a < b ssume u ( α, here α a and α b Then (* becomes u ( α u( α (** Since u is continuous (u5 and monotonic nondecreasing (u3 ith u (0, α α and u (, α -7

8 bout the fuzzy set operations max and min ~ The structure of { ( X,,, P satisfies Involution Commutativity, 3 ssociativity ( C ( C ( C ( C 4 Distributivity 5 Idempotence 6 bsorption 7 bsorption by X and 8 Identity 9 DeMorgan las ut violates a La of contradiction La of excluded middle X Remark on violations: Since a fuzzy set has no sharp boundary, and neither has, and ould overlap, so Hoever, the overlap is alays limited, since u ( u, u min lso, does not cover X, but u ( u, u max 05-8 ~ lternative Operators on P ( X Probabilistic operator Def: Remark: Intersection Union { X },, u u u( x u, x X ˆ, + u + ˆ u + ˆ u( x + u( x u( x u, x X Probabilistic union and intersection can be deduced by the Dubois and Prade class of fuzzy set operation, since as α, a + b ab min( α a + b ab 0 a a + b ab max( α ~ { ( X, + ˆ,, a b a b max( α P satisfies only Commutativity associativity identity DeMorgan's la, +ˆ X X (absorption by and X

9 old operator ~ ~ ~ Comparison on { P ( X,,,, { P ( X, + ˆ,,, { P ( X,,, Def: Intersection Union ( 0, + max, x X ( ( min, + x, x X In general, + ˆ max( 0, + ( x min(, max(, + ( x (, + min Remark: old union and intersection can be deduced by the Yager class operator, since as, a min(,( a + b min(, a + [ ] min, (( a + ( min,( ~ { ( X,,, ( a + ( max(0, a + b P satisfies only Commutativity associativity identity DeMorgan's la, X X (absorption by and X X, (la of exclude middle bout the standard operators > Union > Intersection {, ( } max x x X 3> Complement {, ( } min x x X, x X -9

10 Example: ges Infant dult Young Old Comment: From the note, they perform identically to the corresponding crisp set operators hen {0,} is instead of [0,] The max and min operations are additional significant in that they constitute the only fuzzy union and intersection operators that are continuous and idempotent 3 oth of these operations are very simple 4 They are good generalizations of the classical crisp set operators Young Old Young Old lot of applications are baseing on this pair of operations Note: When e take {0,} instead of [0,], then { } X, max{, } x X { x x or x } { } X, min{, } x X { x x and x } 3 { x x X and x } -0