Fuzzy Sets

Abstract

In this chapter we have collected together the basic ideas from fuzzy sets and fuzzy functions needed for the book. Any reader familiar with fuzzy sets, fuzzy numbers, the extension principle, α-cuts, interval arithmetic, and fuzzy functions may go on and have a look at Sections 2.5 through 2.10. In Section 2.5 we discuss our method of handling the maximun/minimum of a fuzzy set to be used in Chapter 9 and in Section 2.6 we present a method of ordering a finite set of fuzzy numbers from smallest to largest to be employed in Chapters 5–7. Section 2.7 will be used starting in Chapter 3 where we substitute fuzzy numbers for probabilities in discrete probability distributions. Section 2.8 is important starting in Chapter 4 where we show how to obtain fuzzy numbers for uncertain parameters in probability density (mass) functions using a set of confidence intervals. In Section 2.9 we show numerical procedures for computing α-cuts of fuzzy probabilities which will be used throughout the book. Finally, in Section 2.10, we discuss our methods of obtaining the figures for fuzzy probabilities used throughout the book. A good general reference for fuzzy sets and fuzzy logic is [4] and [17].