Approximate Equalities on Rough Intuitionistic Fuzzy Sets and an Analysis of Approximate Equalities
This work is incremental, extending existing approximate equality concepts to a more complex fuzzy set framework for potential applications in areas involving user knowledge and uncertainty.
The paper tackles the problem of defining approximate equalities for sets by extending previous work to rough intuitionistic fuzzy sets, introducing new concepts and studying their properties, with real-life examples provided for application.
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.