Chapter 2 The Operation of Fuzzy Set

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1 Chapter 2 The Operation of Fuzzy Set

2 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

3 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,]

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

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

6 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

7 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

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

9 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

10 3 Drastic sum 4 Hamacher s sum others for, 0 when, 0 when,, X 0, 2, + γ γ γ X Other union operations

11 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.

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

13 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

14 3 Drastic product 4 Hamacher s product Other intersection operations <, when 0, when, when, 0, + + γ γ γ

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

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

17 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.}

18 Other operations in fuzzy set! Simple disjunctive sum Set Set Set Fig 2. Eample of simple disjunctive sum

19 Other operations in fuzzy set! Eclusive or disjoint sum Δ Set Set Set shaded area Fig 2.2 Eample of disjoint sum eclusive OR sum

20 Other operations in fuzzy set! Eclusive or disjoint sum Δ Set Set Set shaded area {, 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.} Fig 2.2 Eample of disjoint sum eclusive OR sum

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

22 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}

23 Other operations in fuzzy set! Simple difference2 Set Set Simple difference - : shaded area Fig 2.4 simple difference

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

25 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,

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

27 Distance in fuzzy set! Euclidean distance e! Minkowski distance n i e 2, ] [,,, / w d w X w w , e

28 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 ]

29 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

30 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

31 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.

32 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 :

Chapter 2: FUZZY SETS

Ch.2: Fuzzy sets 1 Chapter 2: FUZZY SETS Introduction (2.1) Basic Definitions &Terminology (2.2) Set-theoretic Operations (2.3) Membership Function (MF) Formulation & Parameterization (2.4) Complement