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

This ebook is to be the start line for any curriculum in fuzzy platforms in fields like desktop technological know-how, arithmetic, business/economics and engineering. It covers the fundamentals resulting in: fuzzy clustering, fuzzy development reputation, fuzzy database, fuzzy photograph processing, tender computing, fuzzy purposes in operations learn, fuzzy determination making, fuzzy rule established platforms, fuzzy structures modeling, fuzzy arithmetic. it isn't a publication designed for researchers - it really is the place you actually study the "basics" wanted for any of the above-mentioned applications.

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**Sample text**

11. 6. 1 10. Give an example of a continuous fuzzy subset A of R where A[ a] is not an interval for all 0 :::; a :::; 1. 11. How would you define, if possible, a-cuts of type 2, or level 2, fuzzy sets? 12. 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.

F. ((Tb,Cb), (T* ,C*)). 2. 8. a. Find B = A1 n A 2 n A3 using T = Tm,Tp, nand T*. b. Is there any way to control for sp(B)? , B =f:- "¢. c. Find C = A1 U A2 U A3 using C = Cm, Cp, Cb and C*. d. Is there any way to control for sp(C)? Do all the C give the same sp(C) or can you choose a unique C to minimize sp(B)? e. Repeat parts a through d using only A1 and A2 . 3. 1, X = {0, 1, 2, 3, 4, 5}, the values give the membership values for fuzzy sets A 1 , A 2 , A 3 , A 4 at x in X. a. Find B = A1 n A2 n A3 T minimizes sp(B)?

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. Do any of the above (a, b, c) give type 2 fuzzy sets? 11. Give other examples of fuzzy sets besides "young", "fast cars", "smart", ... given in the text. 12. Can we use the following functions for i(a, b)? a. 5}. b. max {O , a+ b +2 ab- 1} . c. 1- J(l- a)2 + (1- b)2- (1- a)2(1- b)2. 13. Can we use the following functions for u(a, b)? a. b. min{1, a+ b + 2ab}. c.