On the approximate equalities of multigranular rough sets and approximate reasoning

Rough sets have been a successful model to capture impreciseness in data. The basic notion rough set introduced by Pawlak is a single granulation model from the granular computing point of view. Recently, this has been extended to 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. In this article, we introduce the concepts of multigranular approximate equalities and establish their properties. Also, the replacement properties, which are obtained by interchanging the bottom equalities with the top equalities, have been established. We provide a real life example for both types of multigranulation and illustrate the interpretation of the approximate equalities through the example.