Basic Notions of the Language of Fuzzy Sets

Abstract

The purpose of this chapter is to present the basics of fuzzy sets. We start with defining the very concept of a fuzzy set, discuss simple examples, and emphasize that many-valued logic is a suitable logical basis for fuzzy sets. Further, the standard approach to operations on fuzzy sets is described. Useful characteristics of a fuzzy set are defined and studied in Section 2.3, including questions of convexity and fuzziness measures. We also discuss the decomposition property of a fuzzy set and the resulting maps of fuzzy sets. A flexible approach to operations on fuzzy sets involving arbitrary negations and triangular norms and conorms is presented in Section 2.4. Implication operators, too, are briefly analyzed therein. The subject of the last section are two interrelated concepts, namely fuzzy numbers and linguistic variables. Fuzzy numbers form a tool for modeling imprecise numerical data. We will define basic types of fuzzy numbers. Second, the extension principle, arithmetic operations on and inequalities between fuzzy numbers will be discussed. Finally, we will move on to the question of linguistic variables, i.e. variables attaining linguistic values interpreted by means of fuzzy numbers.