By Li-Xin Wang
Provides a finished, self-tutorial path in fuzzy good judgment and its expanding position up to the mark theory. The ebook solutions key questions about fuzzy structures and fuzzy keep watch over. It introduces simple suggestions resembling fuzzy units, fuzzy union, fuzzy intersection and fuzzy supplement. find out about fuzzy family, approximate reasoning, fuzzy rule bases, fuzzy inference engines, and several other equipment for designing fuzzy systems. For specialist engineers and scholars using the foundations of fuzzy good judgment to paintings or learn up to speed concept.
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Additional info for A Course in Fuzzy Systems and Control
1. 1), c [ p (x)] ~ = 1- pA(x). In order for the function c to be qualified as a complement, it should satisfy at least the following two requirements: Axiom cl. c(0) = 1 and c(1) = 0 (boundary condition). Axiom c2. For all a,b E [0, 11, if a < b, then c(a) 2 c(b) (nonincreasing condition), where (and throughout this chapter) a and b denote membership functions of some fuzzy sets, say, a = ,UA(X) and b = p~ (x). Axiom c l shows that if an element belongs to a fuzzy set to degree zero (one), then it should belong to the complement of this fuzzy set to degree one (zero).
Axiom s2. s(a, b ) = s(b, a) (commutative condition). Axiom s3. If a 5 a' and b 5 b', then s(a, b) 5 s(al, b') (nondecreasing condition). Axiom s4. s(s(a, b ) , c) = s(a, s(b, c)) (associative condition). Axiom s l indicates what an union function should be in extreme cases. Axiom s2 insures that the order in which the fuzzy sets are combined has no influence on the result. Axiom s3 shows a natural requirement for union: an increase in membership values in the two fuzzy sets should result in an increase in membership value in the union of the two fuzzy sets.
5) satisfies Axioms c l and c2. Fig. 1 illustrates this class of fuzzy complements for different values of A. 1). Another type of fuzzy complement is the Yager class (Yager ) defined by where w E (0, oo). For each value of w ,we obtain a particular fuzzy complement. 6) satisfies Axioms c l and c2. Fig. 2 illustrates the Yager class of fuzzy complements for different values of w. 1). 2 Fuzzy Union-The S-Norms Let s : [O, 11x [O,1] + [0,1]be a mapping that transforms the membership functions of fuzzy sets A and B into the membership function of the union of A and B, that 36 Further Operations on Fuzzy Sets Ch.