Chapter 2 - Basic Notions in Fuzzy Set Theory

Abstract

This chapter reviews fuzzy set representations, fuzzy set connectives (complementation, intersection and union), implication operators, arithmetic operations, ranking of fuzzy numbers, and fuzzy relation equations. The theory of fuzzy sets may be viewed as an attempt at developing a body of concepts and techniques for dealing in a systematic way with a type of imprecision that arises when the boundaries of a class of objects are not sharply defined. Informally, a fuzzy set may be regarded as a class in which there is a graduality of progression from membership to nonmembership or, more precisely, in which an object may have a grade of membership intermediate between unity (full membership) and zero (nonmembership). A fuzzy set is generally assumed to be imbedded in a non-fuzzy universe of discourse, which may be any collection of objects, concepts, or mathematical constructs. For example, a universe of discourse, U, may be the set of all real numbers; the set of integers 0, 1, 2, … ,100; the set of all residents in a city; the set of all students in a course; the set of objects in a room; and the set of all names in a telephone directory. Universes of discourse are usually denoted by the symbols U, V, W, … , with or without subscripts and/or superscripts. A fuzzy set in U or, equivalently, a fuzzy subset of U, is usually denoted by one of the uppercase symbols A, B, C, D, E, F, G, H, with or without subscripts and/or superscripts.