Covering Based Pessimistic Multigranular Approximate Rough Equalities and Their Properties

Rough set model introduced by Pawlak in 1982 was dependent upon equivalence relations (equivalently on partitions), which had restricted its application due to its stringent requirements. The notion of covering is an extension of that of a partition and is generated by relations much less restricted than equivalence relations. Several covering based rough sets are found in literature. As the notion of rough sets, basic or otherwise are unigranular from the granular computing point of view, in order to handle more than one granular structures on a universe simultaneously, optimistic and pessimistic multigranular computing were introduced by Qian et al in 2006 and 2010 respectively. Combining the two concepts of covering and multigranulation, covering based multigranular models were introduced by Liu et al in 2012. The notion of mathematical equality of concepts is too stringent and less applicable in real life situations. In order to incorporate human knowledge into it, four types of approximate equalities basing upon rough sets were introduced by Novotny and Pawlak in 1985 and by Tripathy in 2011. In this paper, we study the covering based pessimistic multigranular approximate rough equalities and establish several of their properties and provide suitable examples for illustration and in constructing counter examples in the proofs. This is an attempt to generalize the notion of approximate equalities by one more level in order to extend their applicability.