Foundations of Fuzzy Set Theory

Abstract

In classical set theory, referred to as crisp set theory to distinguish it from its generalization to fuzzy set theory, an object is either completely in or completely outside of a set. In the former case, the degree of membership of the object is designated as 1 and as 0 in the latter case. Equivalently, the range of the characteristic function of a crisp set consists of the two values 0 and 1. A fuzzy set is a generalization of a crisp set that allows objects to be partially in a set. The membership function of a fuzzy set provides a degree of membership that can range from 0 to 1. The more the object belongs to the fuzzy set, the higher the degree of membership. This chapter briefly presents the notation and terminology of fuzzy set theory that will be used throughout this book. A thorough introduction to fuzzy set theory may be found in a number of books including [61, 123, 263, 246, 183].