An Introduction to Fuzzy Logic and Fuzzy Sets by Professor James J. Buckley, Professor Esfandiar Eslami

By Professor James J. Buckley, Professor Esfandiar Eslami (auth.)

This booklet is to be the start line for any curriculum in fuzzy platforms in fields like computing device technology, arithmetic, business/economics and engineering. It covers the fundamentals resulting in: fuzzy clustering, fuzzy development reputation, fuzzy database, fuzzy photo processing, tender computing, fuzzy purposes in operations learn, fuzzy choice making, fuzzy rule dependent structures, fuzzy structures modeling, fuzzy arithmetic. it isn't a ebook designed for researchers - it truly is the place you actually study the "basics" wanted for any of the above-mentioned applications.

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To do this for a sequence An, each An in Fo(R), we need a metric D(A, B), A, B in F 0 (R). We now present two metrics for F 0 (R). You are asked to investigate other possible metrics in the exercises. Given A,B in Fo(R), set A[a] = [a1(a),a2(a)],B[a] = [b1(a),b2(a)],O::; a::; 1. Define L(a) = la1(a)- b1(a)I,R(a) = la2(a)- b2(a)l. Then D(A, B)= max{max(L(a), R(a))IO::; a::; 1}. 80) Since L(a) and R(a) are continuous we used max instead of sup. This D is a metric. 12. Then a 1 (a) = 1 +a, a 2(a) = 4- 2a, b1 (a) = 1 + 2a, b2(a) = 4- a.

Using i(a, b)= ab, u(a, b)= a+b-ab we may show that De Morgan's law (AUB)c = AcnF holds. J), a~ A(x), b = B(x), and 1 - u(a, b) = 1 - (a+ b- ab). Next evaluate A n B to be i(1- a, 1- b)= (1- a)(1- b). We see that 1- (a+ b- ab) = (1- a)(1- b) so that this De Morgan law is true for i(a, b)= ab, u(a, b) =a+ b- ab. A fuzzy subset of X x Y is called a fuzzy relation. 33) is called a type 1 fuzzy matrix. A type 1 fuzzy matrix has all its elements in [0, 1]. The elements in the fuzzy matrix are all the membership values of the fuzzy relation.

4. 2. We write T = (afb, cfd) for a trapezoidal fuzzy number. 5. 5. We will usually be using triangular ( shaped) and trapezoidal (shaped) fuzzy numbers in this book.

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