Exploring the Fuzzy Frontier: Unraveling the Mysteries of Fuzzy Functions

Abstract

In numerous real-world applications, such as sensor data, the data values are vaguely specified. Fuzzy set theory has been proposed to address such ambiguity by generalizing the concept of set membership. In a Fuzzy Set (FS), each element is essentially associated with a point-value chosen from the unit interval, which is known as the grade of membership in the set. A Vague Set (VS) and an Intuitive Fuzzy Set (IFS) are additional generalizations of a Fuzzy Set (FS). In lieu of point-based membership, interval-based membership is utilized in VSs. Membership in VSs based on intervals is more expressive in capturing imprecise data. In the literature, IFSSs and VSs are considered equivalent in the sense that an IFS is isomorphic with a VS. In addition, as a result of this equivalence and the fact that IFSs were formerly known as a tradition, the unique and fascinating features of VSs for handling imprecise data are largely disregarded. In this paper, we compare VSs and IFSs based on their algebraic properties, graphical representations, and practical applications. Here, we present the idea of a fuzzy function. Crisp functions with fuzzy constraints and fuzzifying functions make up fuzzy functions. To locate the highest possible value within the fuzzy domain of the crisp function, we also present and use the techniques of maximizing and reducing sets. Here, we present the idea of a fuzzy function. Crisp functions with a fuzzy constraint are the building blocks of fuzzy functions, together with the fuzzifying function. In order to locate the highest possible value within the fuzzy domain of the crisp function, we additionally introduce and use the concepts of a maximizing set and a minimizing set.