Chapter 2 The Operation of Fuzzy Set

2.1 Standard operations of fuzzy set Complement set Union A  B Intersection A  B difference between characteristics of crisp fuzzy set operator law of contradiction law of excluded middle

(1) Involution (2) Commutativity A  B = B  A A  B = B  A (3) Associativity (A  B)  C = A  (B  C) (A  B)  C = A  (B  C) (4) Distributivity A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) (5) Idempotency A  A = A A  A = A (6) Absorption A  (A  B) = A A  (A  B) = A (7) Absorption by X and  A  X = X A   =  (8) Identity A   = A A  X = A (9) De Morgan ’ s law (10) Equivalence formula (11) Symmetrical difference formula Table 2.1 Characteristics of standard fuzzy set operators

2.2 Fuzzy complement Requirements for complement function Complement function C: [0,1]  [0,1] (Axiom C1)C(0) = 1, C(1) = 0 (boundary condition) (Axiom C2)a,b  [0,1] if a  b, then C(a)  C(b) (monotonic non-increasing) (Axiom C3) C is a continuous function. (Axiom C4) C is involutive. C(C(a)) = a for all a  [0,1]

2.2 Fuzzy complement Example of complement function(1) C(a) = 1 - a a 1 C(a) 1 Fig 2.1 Standard complement set function

2.2 Fuzzy complement Example of complement function(2) standard complement set function x 1 1 x 1 1

a 1 C(a) 1 t 2.2 Fuzzy complement Example of complement function(3) It does not hold C3 and C4

C(a) 2.2 Fuzzy complement Example of complement function(4) Continuous fuzzy complement function C(a) = 1/2(1+cos  a)

C w (a) w=0.5 w=1 w=2 w=5 a 2.2 Fuzzy complement Example of complement function(5) Yager complement function

2.2 Fuzzy complement Fuzzy Partition (1) (2) (3)

2.3 Fuzzy union Axioms for union function U : [0,1]  [0,1]  [0,1]  A  B (x) = U[  A (x),  B (x)] (Axiom U1) U(0,0) = 0, U(0,1) = 1, U(1,0) = 1, U(1,1) = 1 (Axiom U2) U(a,b) = U(b,a) (Commutativity) (Axiom U3) If a  a’ and b  b’, U(a, b)  U(a’, b’) Function U is a monotonic function. (Axiom U4) U(U(a, b), c) = U(a, U(b, c)) (Associativity) (Axiom U5) Function U is continuous. (Axiom U6) U(a, a) = a (idempotency)

A 1 X B 1 X ABAB 1 X Fig 2.6 Visualization of standard union operation 2.3 Fuzzy union Examples of union function U[  A (x),  B (x)] = Max[  A (x),  B (x)], or  A  B (x) = Max[  A (x),  B (x)]

2.3 Fuzzy union Yager’s union function :holds all axioms except U6.

1) Probabilistic sum (Algebraic sum) commutativity, associativity, identity and De Morgan’s law 2) Bounded sum A  B (Bold union) Commutativity, associativity, identity, and De Morgan’s Law not idempotency, distributivity and absorption Other union operations

3) Drastic sum A B 4) Hamacher’s sum A  B   Other union operations  

I:[0,1]  [0,1]  [0,1] 2.4 Fuzzy intersection Axioms for intersection function (Axiom I1) I(1, 1) = 1, I(1, 0) = 0, I(0, 1) = 0, I(0, 0) = 0 (Axiom I2) I(a, b) = I(b, a), Commutativity holds. (Axiom I3) If a  a’ and b  b’, I(a, b)  I(a’, b’), Function I is a monotonic function. (Axiom I4) I(I(a, b), c) = I(a, I(b, c)), Associativity holds. (Axiom I5) I is a continuous function (Axiom I6) I(a, a) = a, I is idempotency.

ABAB 1 X I[  A (x),  B (x)] = Min[  A (x),  B (x)], or  A  B (x) = Min[  A (x),  B (x)] 2.4 Fuzzy intersection Examples of intersection standard fuzzy intersection

2.4 Fuzzy intersection Yager intersection function

1) Algebraic product (Probabilistic product)  x  X,  A  B (x) =  A (x)   B (x) commutativity, associativity, identity and De Morgan’s law 2) Bounded product (Bold intersection) commutativity, associativity, identity, and De Morgan’s Law not idempotency, distributivity and absorption Other intersection operations   

3) Drastic product A B 4) Hamacher’s product A  B Other intersection operations    

A B Fig 2.10 Disjunctive sum of two crisp sets 2.5 Other operations in fuzzy set Disjunctive sum

2.5 Other operations in fuzzy set Simple disjunctive sum

2.5 Other operations in fuzzy set Simple disjunctive sum(2) ex)

Fig 2.11 Example of simple disjunctive sum 2.5 Other operations in fuzzy set Simple disjunctive sum(3)

2.5 Other operations in fuzzy set (Exclusive or) disjoint sum Fig 2.12 Example of disjoint sum (exclusive OR sum)

2.5 Other operations in fuzzy set (Exclusive or) disjoint sum Fig 2.12 Example of disjoint sum (exclusive OR sum) A = {(x 1, 0.2), (x 2, 0.7), (x 3, 1), (x 4, 0)} B = {(x 1, 0.5), (x 2, 0.3), (x 3, 1), (x 4, 0.1)} A △ B = {(x 1, 0.3), (x 2, 0.4), (x 3, 0), (x 4, 0.1)}

2.5 Other operations in fuzzy set Difference in fuzzy set Difference in crisp set A B Fig 2.13 difference A – B

2.5 Other operations in fuzzy set Simple difference ex)

A B x1x1 x2x2 x3x3 x4x4 Set A Set B Simple difference A-B : shaded area Fig 2.14 simple difference A – B 2.5 Other operations in fuzzy set Simple difference(2)

x1x1 x2x2 x3x3 x4x4 Set A Set B Bounded difference : shaded area A B 0.4 Fig 2.15 bounded difference A  B 2.5 Other operations in fuzzy set  A  B (x) = Max[0,  A (x) -  B (x)] Bounded difference A  B = {(x 1, 0), (x 2, 0.4), (x 3, 0), (x 4, 0)}

2.5.3 Distance in fuzzy set Hamming distance d(A, B) = 1. d(A, B)  0 2. d(A, B) = d(B, A) 3. d(A, C)  d(A, B) + d(B, C) 4. d(A, A) = 0 ex)A = {(x 1, 0.4), (x 2, 0.8), (x 3, 1), (x 4, 0)} B = {(x 1, 0.4), (x 2, 0.3), (x 3, 0), (x 4, 0)} d(A, B) = |0| + |0.5| + |1| + |0| = 1.5

A x 1  A (x) B x 1  B (x) B A x 1  A (x) B A x 1  B (x)  A (x) distance between A, B difference A- B Distance in fuzzy set Hamming distance : distance and difference of fuzzy set

2.5.3 Distance in fuzzy set Euclidean distance ex) Minkowski distance

2.5.4 Cartesian product of fuzzy set Power of fuzzy set Cartesian product

2.6 t-norms and t-conorms Definitions for t-norms and t-conorms t-norm T : [0,1]  [0,1]  [0,1]  x, y, x’, y’, z  [0,1] i) T(x, 0) = 0, T(x, 1) = x: boundary condition ii) T(x, y) = T(y, x): commutativity iii) (x  x’, y  y’)  T(x, y)  T(x’, y’): monotonicity iv) T(T(x, y), z) = T(x, T(y, z)): associativity 1) intersection operator (  ) 2) algebraic product operator (  ) 3) bounded product operator ( ) 4) drastic product operator ( )   

2.6 t-norms and t-conorms t-conorm (s-norm) T : [0,1]  [0,1]  [0,1]  x, y, x’, y’, z  [0,1] i) T(x, 0) = 0, T(x, 1) = 1: boundary condition ii) T(x, y) = T(y, x): commutativity iii) (x  x’, y  y’)  T(x, y)  T(x’, y’): monotonicity iv) T(T(x, y), z) = T(x, T(y, z)): associativity 1) union operator (  ) 2) algebraic sum operator ( ) 3) bounded sum operator (  ) 4) drastic sum operator ( ) 5) disjoint sum operator (  )  

2.6 t-norms and t-conorms Ex) a)  : minimum Instead of *, if  is applied x  1 = x Since this operator meets the previous conditions, it is a t-norm. b)  : maximum If  is applied instead of *, x  0 = x then this becomes a t-conorm.

2.6 t-norms and t-conorms Duality of t-norms and t-conorms