CHAPTER 2

BASIC CONCEPTS OF FUZZY SETS

This book shows how fuzzy logic can be used for identification and control of dynamic systems. The foundation of fuzzy logic is the fuzzy set. The concept of the fuzzy set was first introduced by Zadeh in [5,6]. The fuzzy set is a generalization of the conventional, or crisp, set that is well known to math and engineering students (see, however, even a generalization of the fuzzy set, given in [7]). In this chapter, we give some basic concepts of fuzzy sets that will be useful for the topics covered in this book (i.e., fuzzy sets, universes of discourse, linguistic variables, linguistic values, membership functions, and some associated set-theoretic operations involving them).

2.1 FUZZY SETS

For the purposes of this book, a fuzzy set is a collection of real numbers having partial membership in the set. This is in contrast with conventional, or crisp sets, to which a number can belong or not belong, but not partially belong. For example, consider the set of “all heights of people 6-ft tall or taller.” This is a collection of all real numbers ≥6. It is a crisp set because a number either belongs to this set or does not belong to it. It is impossible for a number to partially belong to the set. Now consider a different kind of set, the set of “heights of tall people.” A height of 7-ft tall is definitely considered tall, a height of 5-ft tall is definitely not considered tall, and a height of 6-ft tall may be considered “kind of tall,” or tall ...