On Approximation of Classifications, Rough Equalities and Rough Equivalences

Abstract

In this chapter we mainly focus on the study of some topological aspects of rough sets and approximations of classifications. The topological classification of rough sets deals with their types. We find out types of intersection and union of rough sets, New concepts of rough equivalence (top, bottom and total) are defined, which capture approximate equality of sets at a higher level than rough equality (top, bottom and total) of sets introduced and studied by Novotny and Pawlak [23,24,25] and is also more realistic. Properties are established when top and bottom rough equalities are interchanged. Also, parallel properties for rough equivalences are established. We study approximation of classifications (introduced and studied by Busse [12]) and find the different types of classifications of an universe completely. We find out properties of rules generated from information systems and observations on the structure of such rules. The algebraic properties which hold for crisp sets and deal with equalities loose their meaning when crisp sets are replaced with rough sets. We analyze the validity of such properties with respect to rough equivalences.