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

This publication is to be the place to begin for any curriculum in fuzzy platforms in fields like laptop technology, arithmetic, business/economics and engineering. It covers the fundamentals resulting in: fuzzy clustering, fuzzy development acceptance, fuzzy database, fuzzy snapshot processing, gentle computing, fuzzy functions in operations study, fuzzy selection making, fuzzy rule dependent structures, fuzzy structures modeling, fuzzy arithmetic. it's not a publication designed for researchers - it truly is the place you actually research the "basics" wanted for any of the above-mentioned applications.

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**Sample text**

2, ... 7. There are different types of fuzzy sets. 31). 31) X was finite but we can also have discrete fuzzy sets for X infinite. Let X= Rand A a fuzzy subset of R so that A(x) "I- 0 only for x in {1, 2, ... , 10}. 34) a discrete fuzzy set, where A(i) = J-Li, 1 ~ i ~ 10. 35) A = {d /-Ll . ' h. J-L~ . ' ... } . epresswn sc ~zop renza At the other extreme of discrete fuzzy sets we have "continuous " fuzzy sets, fuzzy subsets of R whose membership functions are continuous. 1 gives an example of a continuous fuzzy subset of R.

Let C = An B. Clearly, C(x) = 1 if A(x) = B(x) = 1 and C(x) = 0 whenever A(x) or B(x) = 0. 4? The value of C(x) will be a function of the two values A(x) and B(x). Let this unknown function be called i(a, b), for intersection, with a, b in [0, 1]. Here we let a = A(x), b = B(x) so that C(x) = i(a, b). This function i(a, b) must have the properties: (1) 0:::; a, b:::; 1 and i(a, b) is in [0, 1]; (2) i(1, 1) = 1; and (3) i(O, 1) = i(1, 0) = i(O, 0) = 0. 21) for all x in X. 21) defines the intersection of two fuzzy sets and B by defining the membership function of C.

17) Let us show how these may be proven using membership functions. 13). Let C = (An B)e and D = Ae U Be and we show that C(x) = D(x) for all x in X. Now C(x) = 1 if (An B)(x) = 0 and C(x) = 0 for (An B)(x) = 1. 18) 0, if A(x) and B(x) = 1. Also, D(x) = 1 if Ae(x) or Be(x) = 1 and D(x) = 0 for Ae(x) and Be(x) = 0. Hence D(x) = { 1, if A(x) or B(x) = 0 0, if A(x) and B(x) = 1. 19), C(x) = D(x), for all x in X, and this De Morgan law holds. 17). Let C =An (AU B). Then C(x) = 1 if A(x) = 1 and (AU B)(x) = 1 and C(x) = 0 if A(x) = 0 or (AU B)(x) = 0.