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§3.3 Fuzzy Relation 71 It’s corresponding fuzzy matrix is as follows. a b c d A a 0.0 0.0 0.8 0.0 b 1.0 0.0 0.0 0.0 c 0.0 0.9 0.0 1.0 d 0.0 0.0 0.0 0.0 Fuzzy relation is mainly useful when expressing knowledge. Generally, the knowledge is composed of rules and facts. A rule can contain the concept of possibility of event b after event a has occurred. For instance, let us assume that set A is a set of events and R is a rule. Then by the rule R, the possibility for the occurrence of event c after event a occurred is 0.8 in the previous fuzzy relation. When crisp relation R represents the relation from crisp sets A to B, its domain and range can be defined as, dom(R) {x | x A, y A, P (x, y) 1} R ran(R) {y | x A, y A, P (x, y) 1} R Definition (Domain and range of fuzzy relation) When fuzzy relation R is defined in crisp sets A and B, the domain and range of this relation are defined as : P (x) max P (x, y) dom(R) R yB P (y) max P (x, y) ran(R) R xA Set A becomes the support of dom(R) and dom(R) A. Set B is the support of ran(R) and ran(R) B. Ƒ 3.3.3 Fuzzy Matrix Given a certain vector, if an element of this vector has its value between 0 and 1, we call this vector a fuzzy vector. Fuzzy matrix is a gathering of 72 3. Fuzzy Relation and Composition such vectors. Given a fuzzy matrix A (a ) and B (b ), we can perform ij ij operations on these fuzzy matrices. (1) Sum A + B Max [a , b ] ij ij (2) Max product AxB AB Max [ Min (a ,b ) ] k ik kj (3) Scalar product OA where 0 dOd 1 Example 3.4 The followings are examples of sum and max product on fuzzy sets A and B. a b c a b c A = a 0.2 0.5 0.0 B = a 1.0 0.1 0.0 b 0.4 1.0 0.1 b 0.0 0.0 0.5 c 0.0 1.0 0.0 c 0.0 1.0 0.1 a b c a b c A + B = a 1.0 0.5 0.0 AxB a 0.2 0.1 0.5 b 0.4 1.0 0.5 b 0.4 0.1 0.5 c 0.0 1.0 0.1 c 0.0 0.0 0.5 Here let's have a closer look at the product A x B of A and B. For instance, in the first row and second column of the matrix C A x B, the value 0.1 (C 0.1) is calculated by applying the Max-Min operation to 12 the values of the first row (0.2, 0.5 and 0.0) of A, and those of the second column (0.1 , 0.0 and 1.0) of B. 0.2 0.5 0.0 0.1 0.0 1.0 Min 0.1 0.0 0.0 0.1 Max In the same manner C13 0.5 is obtained by applying the same procedure of calculation to the first row (0.2, 0.5, 0.0) of A and the third column of B (0.0, 0.5, 0.1). §3.3 Fuzzy Relation 73 0.2 0.5 0.0 0.0 0.5 0.1 Min 0.0 0.5 0.0 0.5 Ƒ Max And for all i and j, if a d b holds, matrix B is bigger than A. ij ij aij d bij AdB Also when A d B for arbitrary fuzzy matrices S and T, the following relation holds from the Max-Product operation. AdB SAdSB,ATdBT Definition (Fuzzy relation matrix) If a fuzzy relation R is given in the form of fuzzy matrix, its elements represent the membership values of this relation. That is, if the matrix is denoted by M , and membership values by R P (i, j), then M (P (i, j)) Ƒ R R R 3.3.4 Operation of Fuzzy Relation We know now a relation is one kind of sets. Therefore we can apply operations of fuzzy set to the relation. We assume R A u B and S A u B. (1) Union relation Union of two relations R and S is defined as follows : (x, y) A u B P (x, y) Max [P (x, y), P (x, y)] RS R S P (x, y) P (x, y) R S We generally use the sign for Max operation. For n relations, we extend it to the following. PRR R R x, y PRi x,y 1 2 3 n R i If expressing the fuzzy relation by fuzzy matrices, i.e. M and M , R S matrix M concerning the union is obtained from the sum of two R S matrices M + M . R S M M + M RS R S (2) Intersection relation The intersection relation R S of set A and B is defined by the following membership function. 74 3. Fuzzy Relation and Composition P (x) =Min [P (x, y), P (x, y)] R S R S = P (x, y) P (x, y) R S The symbol is for the Min operation. In the same manner, the intersection relation for n relations is defined by PR R R R x, y PRix, y 1 2 3 n Ri (3) Complement relation Complement relation R for fuzzy relation R shall be defined by the following membership function. (x, y) A u B P (x, y) 1 - P (x, y) R R Example 3.5 Two fuzzy relation matrices M and M are given. R S M a b c M a b c R S 1 0.3 0.2 1.0 1 0.3 0.0 0.1 2 0.8 1.0 1.0 2 0.1 0.8 1.0 3 0.0 1.0 0.0 3 0.6 0.9 0.3 Fuzzy relation matrices M and M corresponding R S and R S RS R S yield the followings. MRS a b c MRS a b c 1 0.3 0.2 1.0 1 0.3 0.0 0.1 2 0.8 1.0 1.0 2 0.1 0.8 1.0 3 0.6 1.0 0.3 3 0.0 0.9 0.0 Also complement relation of fuzzy relation R shall be MR a b c 1 0.7 0.8 0.0 2 0.2 0.0 0.0 3 1.0 0.0 1.0 (4) Inverse relation When a fuzzy relation R A u B is given, the inverse relation of R-1 is defined by the following membership function. For all (x, y) A u B, P -1 (y, x) P (x, y) R R
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