Introduction

Abstract

Fuzzy theory is firmly grounded in the mathematics of fuzzy sets and fuzzy logic developed by Lofti A. Zadeh in 1965 [166]. Since its introduction, fuzzy set theory has grown to become a major scientific domain. A fuzzy system is referred to as any static or dynamic system which makes use of fuzzy logic and of the corresponding mathematical framework. Such a system allows a gradual and continuous transition, say, from 0 to 1, rather than a crisp and abrupt change between binary values of 0 and 1. It is known that in an ordinary set, an element of the universe either belongs to or does not belong to the set. This shows the membership of an element is crisp. A fuzzy set is a generalization of an ordinary set by allowing a degree of membership for each element, which is a real number on [0, 1]. The membership function of a set maps each element to its degree. Accordingly, fuzzy logic is an extension of ordinary logic, which can represent fuzzy implications. There are a lot of publications to address the details of fuzzy theory, see, e.g., [65][66][67][71][124][168][176].