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

This e-book is to be the start line for any curriculum in fuzzy structures in fields like machine technological know-how, arithmetic, business/economics and engineering. It covers the fundamentals resulting in: fuzzy clustering, fuzzy trend popularity, fuzzy database, fuzzy picture processing, smooth computing, fuzzy purposes in operations learn, fuzzy determination making, fuzzy rule established platforms, fuzzy platforms modeling, fuzzy arithmetic. it isn't a e-book designed for researchers - it really is the place you actually research the "basics" wanted for any of the above-mentioned applications.

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Extra resources for An Introduction to Fuzzy Logic and Fuzzy Sets

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6. Discuss intersection and union for level 2 fuzzy sets. 7. If A::::; Band B::::; C, then does A::::; C hold? 8. Answer the following questions: a. When does X x Y = Y x X? b. If A is crisp subset of X and B,C are crisp subset of Y, does A X (B n C) equal (A X B) n (A X C)? 9. Using i(a, b) =min, show i(A(x), B(x)) ::::; ht(A n B) ::::; max{ht(A), ht(B)}, for all x. 30 CHAPTER 3. FUZZY SETS 10. Discuss the elements in the sets: a. P(P (X)). b. P(F (X)). c. F(P (X)). F(F (X)) gives level 2 fuzzy sets.

How would you define a-cuts of fuzzy subsets of R x R (for example, fuzzy relations, Chapter 7)? 13. Find relationships, if any, between the sets (An B) [a] if we use t-norm Tm, Tp and T* to calculate An B. n, 14. Find relationships, if any, between the sets (AUB)[a] if we use t-conorm Cm, Cb, Cp and C* to find Au B. 15. Determine if the following equations are true or false. All fuzzy sets are fuzzy subsets of the real numbers: a. b. c. 50 CHAPTER 3. 7 Distance Between Fuzzy Sets Let F 0 (R) be all continuous fuzzy subsets of R whose a-cuts are always bounded intervals.

69) for all a in [0, 1]. 69) say that T and C have to be idempotent. The only idempotent t-norm is T m and the only idempotent t-conorm is Cm (see the exercises). So the idempotent laws hold for fuzzy sets only if we use Tm and Cm. Next we come to the law of contradiction A n A c = (/) and the law of excluded middle AU Ac =X. 32)). 4. ALGEBRA OF FUZZY SETS 39 see using Cm that AU Ac -1- X. So these two laws may not hold if we use T m and Cm. We need to investigate if these laws are true or false if we employ Tp and Cp, nand Cb, or T* and C*.

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