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Applic�tions to Classifi cation and Pattern Recognition, Automata and Syste��l.. f_hois_� of Criteria, Masson, Paris, 197 5 . Also English translation of Vol . 1 , Academic, New York, 1975 . 5 . Mizumoto, M . , and K . Tanaka , Some properties of fuzzy sets of type 2, Inf , Control 31, 312-340 (1976) . 6 . Negoita, C . V . , and D. A . Ralescu, Applications of Fuzzy Sets to Systems Analysis , Birkhauser, Basel, 197 5 . 7 . Zadeh, L . A . , Similarity relations and fuzzy orderings , Inf. Sci . �.

U ) ' i . e . , R n n If . • • (42) where µR is the membership function of R . Common examples of (binary) fuzzy relations are: much greater than, resembles , is relevant to, and is close to . For example , if u u (-oo, oo) , the relation 2 1 is close to may be defined by == is clos� to � J U1xU 2 e-a lu1-u 2 II (u1, u 2) == (43) where a is a scale factor . 3 4 0 0 0 0 1 == == (44 ) in which the (i, j)-th element is the value of µ R (u1 , u2 ) for the i-th value of u1 and the j-th value of u 2 • If R is a relation from U to V (or, equivalently, a relation in U x V) and S is a relation from V to W, then the composition of R and S is a fuzzy relation from U to W denoted by R 0 S and defined by R 0 S == J ( (u, v) v µR V uxw A µ5 (v, w))/ (u, w) (45) If U, V , and W are finite sets , then the relation matrix for R 0 S is the max-min product of the relation matrices for R and S .

8/10 + 1/12 + 0. 7/14 + 0. 8/15 + 0. 8/18 + o. 7/21 (70) Thus the arithmetic product of the fuzzy numbers approximately 2 and approxi mately 6 is a fuzzy number given by (70) . More generally, let * be a binary operation defined on U x V with values in W. Thus, if u E U and v E V, then w = u * v, wEW Now suppose that A and B are fuzzy subsets of U and V, respectively, with 41 42 Chapter 2 Basic Notions in Fuzzy Set Theory and A = µ u + •.. + µ u n n 1 1 (71) B = v v + •·· + v v m m 1 1 By using the extension principle under the assumption (67) , the operation * may be extended to fuzzy subsets of U and V by the defining relation A * B= = ( � µiui ) * (� vjvj ) .