CHAPTER 2 - Fuzzy Set Theory for Applications

Abstract

This chapter describes the fuzzy set theory for applications. The two large divisions of crisp set theory are axiomatic set theory and naive set theory. The former is a part of basic mathematical theory and also requires a high level of philosophical thinking. Naive set theory is sufficient here, and this is just an extension of the set theory learned in elementary school. The concept of characteristic functions in naive set theory is necessary. Fuzzy sets are clearly defined by the membership function and there is nothing ambiguous about the definition itself. That is, the logic for the conceptual definition is crisp and absolutely unambiguous. It is clearly defined by an evaluation on X, that is, by the membership function. It is found that when it comes to applications, cases in which X is a finite set are most often considered, but in these instances, the expression for the membership function is mostly written using separators symbols. It can easily be shown that the idempotent, commutative, associative, absorption, distributive, double negative, and de Morgans laws all arise. The only one that does not arise is the law of complements.