Approximate Equalities on Rough Intuitionistic Fuzzy Sets and an Analysis of Approximate Equalities

TL;DR

This paper extends the concept of approximate equalities from rough sets and fuzzy sets to rough intuitionistic fuzzy sets, analyzing their properties and potential real-life applications.

Contribution

It introduces and studies approximate equalities for rough intuitionistic fuzzy sets, expanding the theoretical framework and applications beyond existing rough and fuzzy set models.

Findings

Defined new approximate equalities for rough intuitionistic fuzzy sets

Established properties of these new equalities

Provided real-life examples demonstrating applications

Abstract

In order to involve user knowledge in determining equality of sets, which may not be equal in the mathematical sense, three types of approximate (rough) equalities were introduced by Novotny and Pawlak ([8, 9, 10]). These notions were generalized by Tripathy, Mitra and Ojha ([13]), who introduced the concepts of approximate (rough) equivalences of sets. Rough equivalences capture equality of sets at a higher level than rough equalities. More properties of these concepts were established in [14]. Combining the conditions for the two types of approximate equalities, two more approximate equalities were introduced by Tripathy [12] and a comparative analysis of their relative efficiency was provided. In [15], the four types of approximate equalities were extended by considering rough fuzzy sets instead of only rough sets. In fact the concepts of leveled approximate equalities were introduced and properties were studied. In this paper we proceed further by introducing and studying the approximate equalities based on rough intuitionistic fuzzy sets instead of rough fuzzy sets. That is we introduce the concepts of approximate (rough)equalities of intuitionistic fuzzy sets and study their properties. We provide some real life examples to show the applications of rough equalities of fuzzy sets and rough equalities of intuitionistic fuzzy sets.