On Some Topological Properties of Pessimistic Multigranular Rough Sets

Abstract

Rough set theory was introduced by Pawlak as a model to capture impreciseness in data and since then it has been established to be a very efficient tool for this purpose. The definition of basic rough sets depends upon a single equivalence relation defined on the universe or several equivalence relations taken one each at a time. There have been several extensions to the basic rough sets introduced since then in the literature. From the granular computing point of view, research in classical rough set theory is done by taking a single granulation. It has been extended to multigranular rough set (MGRS) model, where the set approximations are defined by taking multiple equivalence relations on the universe simultaneously. Multigranular rough sets are of two types; namely optimistic MGRS and pessimistic MGRS. Topological properties of rough sets introduced by Pawlak in terms of their types were studied by Tripathy and Mitra to find the types of the union, intersection and complement of such sets. Tripathy and Raghavan have extended the topological properties of basic single granular rough sets to the optimistic MGRS context. Incomplete information systems take care of missing values for items in data tables. MGRS has also been extended to such type of incomplete information systems. In this paper we have carried out the study of topological properties of pessimistic MGRS by finding out the types of the union, intersection and complement of such sets. Also, we have provided proofs and examples to illustrate that the multiple entries in the table can actually occur in practice. Our results hold for both complete and incomplete information systems. The multiple entries in the tables occur due to impreciseness and ambiguity in the information. This is very common in many of the real life situations and needed to be addressed to handle such situations in efficient manner.