On Approximate Equivalences of Multigranular Rough Sets and Approximate Reasoning

The notion of rough sets introduced by Pawlak has been a successful model to capture impreciseness in data and has numerous applications. Since then it has been extended in several ways. The basic rough set introduced by Pawlak is a single granulation model from the granular computing point of view. Recently, this has been extended to two types of multigranular rough set models. Pawlak and Novotny introduced the notions of rough set equalities which is called approximate equalities. These notions of equalities use the user knowledge to decide the equality of sets and hence generate approximate reasoning. However, it was shown by Tripathy et al, even these notions have limited applicability to incorporate user knowledge. So the notion of rough equivalence was introduced by them. The notion of rough equalities in the multigranulation context was introduced and studied. In this article, we introduce the concepts of multigranular rough equivalences and establish their properties. Also, the replacement properties, which are obtained by interchanging the bottom equivalences with the top equivalences, have been established. We provide a real life example for both types of multigranulation, compare the rough multigranular equalities with the rough multigranular equivalences and illustrate the interpretation of the rough equivalences through the example.