Intuitionistic Fuzzy Sets as a Generalization of Fuzzy Sets

Abstract

In the mid-1980s Atanassov introduced the concept of an intuitionistic fuzzy set. Basically, his idea was that unlike the conventional fuzzy sets in which imprecision is just modeled by the membership degree from [0,1], and for which the non-membership degree is just automatically the complementation to 1 of the membership degree, in an intuitionistic fuzzy set both the membership and nonmembership degrees are numbers from [0,1], but their sum is not necessarily 1. Thus, one can express a well known psychological fact that a human being who expresses the degree of membership of an element in a fuzzy set, very often does not express, when asked, the degree of non-membership as the complementation to 1. This idea has led to an interesting theory whose point of departure is such a concept of intuitionistic fuzzy set. In this chapter we give brief introduction to intuitionistic fuzzy sets. After recalling main definitions, concepts, operations and relations over crisp sets, fuzzy sets, and intuitionistic fuzzy sets we discuss interrelationships among the three types of sets. Two geometrical representations of the intuitionistic fuzzy sets, useful in further considerations are discussed. Finally, two approaches of constructing the intuitionistic fuzzy sets from data are presented. First approach is via asking experts. Second one - the automatic, and mathematically justified method to construct the intuitionistic fuzzy sets from data seems to be especially important in the context of analyzing information in big data bases.