Approximate Reasoning through Multigranular Approximate Rough Equalities

The notion of rough set was introduced by Pawlak as an uncertainty based model, which basically depends upon single equivalence relations defined over a universe or a set of equivalence relations, which are not considered simultaneously. Hence, from the granular computing point of view it is unigranular by nature. Qian et al in 2006 and in 2010 introduced two types of multigranular rough sets (MGRS) called the optimistic and pessimistic MGRS respectively. The stringent notion of mathematical equality of sets was extended by introducing a kind of approximate equality, called rough equality by Novotny and Pawlak, which uses basic rough sets. Later three more related types of such approximate equalities have been introduced by Tripathy et al. He has also provided a comparative analysis of these four types of approximate equalities of sets leading to approximate reasoning in real life situations. Two of these four types of approximate equalities; namely the rough equality and rough equivalence have been extended to the context of multigranulations by Tripathy et al very recently. In this paper we carry out this study further by introducing the notion of approximate rough equalities for multigranulations and establish their properties. We use a real life example to illustrate the results in the paper and also to construct examples in support of some parts of the properties.