Chapter 2 The Operation of Fuzzy Set

Chapter 2 The Operation of Fuzzy Set

Standard operations of Fuzzy Set! Complement set! Union Ma[, ]! Intersection Min[, ]! difference between characteristics of crisp fuzzy set operator n law of contradiction n law of ecluded middle X

Fuzzy complement! Requirements for complement function n Complement function C: [0,] [0,] C iom C C0, C 0 boundary condition iom C2 a,b [0,] if a < b, then Ca Cb monotonic non-increasing iom C3 C is a continuous function. iom C4 C is involutive. CCa a for all a [0,]

Fuzzy complement! Eample of complement function Ca Ca - a Fig 2. Standard complement set function a

Fuzzy complement! Eample of complement function n standard complement set function

Fuzzy complement! Eample of complement function3 Ca C a 0 for a t for a > t It does not hold C3 and C4 t a

Fuzzy union! ioms for union function U : [0,] [0,] [0,] U[, ] iom U U0,0 0, U0,, U,0, U, iom U2 Ua,b Ub,a Commutativity iom U3 If a a and b b, Ua, b Ua, b Function U is a monotonic function. iom U4 UUa, b, c Ua, Ub, c ssociativity iom U5 Function U is continuous. iom U6 Ua, a a idempotency

Fuzzy union! Eamples of union function U[, ] Ma[, ], or Ma[, ] X X X Fig 2.6 Visualization of standard union operation

Other union operations Probabilistic sum +ˆ lgebraic sum + X, ˆ + n n commutativity, associativity, identity and De Morgan s law +ˆ X X 2 ounded sum old union X, Min[, + ] n n n Commutativity, associativity, identity, and De Morgan s Law X X, not idempotency, distributivity and absorption X

3 Drastic sum 4 Hamacher s sum others for, 0 when, 0 when,, X 0, 2, + γ γ γ X Other union operations

Fuzzy intersection! ioms for intersection function I:[0,] [0,] [0,] I[, ] iom I I,, I, 0 0, I0, 0, I0, 0 0 iom I2 Ia, b Ib, a, Commutativity holds. iom I3 If a a and b b, Ia, b Ia, b, Function I is a monotonic function. iom I4 IIa, b, c Ia, Ib, c, ssociativity holds. iom I5 I is a continuous function iom I6 Ia, a a, I is idempotency.

Fuzzy intersection! Eamples of intersection n standard fuzzy intersection I[, ] Min[, ], or Min[, ] X

Other intersection operations lgebraic product Probabilistic product X, n commutativity, associativity, identity and De Morgan s law 2 ounded product old intersection X, Ma[0, + ] n n n commutativity, associativity, identity, and De Morgan s Law, not idempotency, distributivity and absorption

3 Drastic product 4 Hamacher s product Other intersection operations <, when 0, when, when, 0, + + γ γ γ

Other operations in fuzzy set! Disjunctive sum Fig 2.0 Disjunctive sum of two crisp sets

Other operations in fuzzy set! Simple disjunctive sum -, - Min[, ] Min[, ], then Ma{ Min[, ], Min[, ]}

Other operations in fuzzy set! Simple disjunctive sum2 e {, 0.2, 2, 0.7, 3,, 4, 0} {, 0.5, 2, 0.3, 3,, 4, 0.} {, 0.8, 2, 0.3, 3, 0, 4, } {, 0.5, 2, 0.7, 3, 0, 4, 0.9} {, 0.2, 2, 0.7, 3, 0, 4, 0} {, 0.5, 2, 0.3, 3, 0, 4, 0.} {, 0.5, 2, 0.7, 3, 0, 4, 0.}

Other operations in fuzzy set! Simple disjunctive sum3.0 0.9.0 Set Set Set 0.8 0.7 0.6 0.7 0.5 0.4 0.5 0.3 0.2 0. 0.2 0.3 0. 0 2 3 4 Fig 2. Eample of simple disjunctive sum

Other operations in fuzzy set! Eclusive or disjoint sum Δ.0 0.9.0 Set Set Set shaded area 0.8 0.7 0.6 0.7 0.5 0.4 0.5 0.3 0.2 0. 0.2 0.3 0. 0 2 3 4 Fig 2.2 Eample of disjoint sum eclusive OR sum

Other operations in fuzzy set! Eclusive or disjoint sum Δ.0 0.9 0.8.0 Set Set Set shaded area 0.7 0.6 0.5 0.4 0.5 0.7 {, 0.2, 2, 0.7, 3,, 4, 0} {, 0.5, 2, 0.3, 3,, 4, 0.} {, 0.3, 2, 0.4, 3, 0, 4, 0.} 0.3 0.2 0. 0.2 0.3 0. 0 2 3 4 Fig 2.2 Eample of disjoint sum eclusive OR sum

Other operations in fuzzy set! Difference in fuzzy set n Difference in crisp set Fig 2.3 difference

Other operations in fuzzy set! Simple difference Min[, ] e {, 0.2, 2, 0.7, 3,, 4, 0} {, 0.5, 2, 0.3, 3,, 4, 0.} {, 0.5, 2, 0.7, 3, 0, 4, 0.9} {, 0.2, 2, 0.7, 3, 0, 4, 0}

Other operations in fuzzy set! Simple difference2 Set 0.7 0.5 0.3 0.7 Set Simple difference - : shaded area 0.2 0. 0.2 2 3 4 Fig 2.4 simple difference

Other operations in fuzzy set! ounded difference θ Ma[0, - ] 0.7 0.5 0.3 0.2 0. 0.4 Set Set ounded difference : shaded area θ {, 0, 2, 0.4, 3, 0, 4, 0} 2 3 4 Fig 2.5 bounded difference θ

Distance in fuzzy set! Hamming distance d, n i, X i i i. d, 0 2. d, d, 3. d, C d, + d, C 4. d, 0 e {, 0.4, 2, 0.8, 3,, 4, 0} {, 0.4, 2, 0.3, 3, 0, 4, 0} d, 0 + 0.5 + + 0.5

Distance in fuzzy set! Hamming distance : distance and difference of fuzzy set distance between, difference -

Distance in fuzzy set! Euclidean distance e! Minkowski distance n i e 2, ] [,,, / w d w X w w.2.25 0 0.5 0, 2 2 2 2 + + + e

Cartesian product of fuzzy set! Power of fuzzy set 2 2 [ ], m m [ ], X X! Cartesian product,,, as membership functions of, 2,, n 2,, n n for, 2 2 n., 2,, n Min[,, n 2 n n ]

t-norms and t-conorms Definitions for t-norms and t-conorms! t-norm T : [0,] [0,] [0,], y,, y, z [0,] i T, 0 0, T, : boundary condition ii T, y Ty, : commutativity iii, y y T, y T, y : monotonicity iv TT, y, z T, Ty, z : associativity intersection operator 2 algebraic product operator 3 bounded product operator 4 drastic product operator

t-norms and t-conorms! t-conorm s-norm T : [0,] [0,] [0,], y,, y, z [0,] i T, 0 0, T, : boundary condition ii T, y Ty, : commutativity iii, y y T, y T, y : monotonicity iv TT, y, z T, Ty, z : associativity union operator 2 algebraic sum operator +ˆ 3 bounded sum operator 4 drastic sum operator 5 disjoint sum operator Δ

t-norms and t-conorms E a : minimum Instead of *, if is applied Since this operator meets the previous conditions, it is a t-norm. b : maimum If is applied instead of *, 0 then this becomes a t-conorm.

t-norms and t-conorms! Duality of t-norms and t-conorms Law by De Morgane's T T, T T, T, y y y y y y y y y y conorm t y norm t y : T :