Some topological properties of rough sets and their applications

Abstract

Properties of approximations of sets establish that lower approximation of union of sets is not equal to the union of their lower approximations in general and a similar result holds for upper approximation of intersection of sets. This confirms to the observation that there is a loss of information in a distributed knowledge base than in an integrated one. We obtain necessary and sufficient conditions for equalities to hold in the above two properties. We find that there are ambiguities in types of union and intersection of rough sets and establish theorems to reduce these ambiguities. Novotny and Pawlak introduced three (bottom, top and total) types of rough equalities, which show that comparison of domains depends upon our knowledge about the universe. We obtain suitable conditions under which the top and bottom rough equalities can be interchanged in these properties. Example of a distributed database of employees is used throughout.