Rough Sets and Zadeh's Extension Principles

The notion of a rough set was originally proposed by Pawlak (1982). Later on, there is a fast growing interest in this theory. In this paper we present a more general approach to the generalization of rough sets. Specifically, generalized formulation has been studied by using a binary relation on two universes without any restriction on the cardinality. The algebraic properties of generalized rough sets are given, and the extension principle for crisp sets is explained as the upper approximations of rough sets. The relationship between the extension principle for fuzzy sets and the upper approximations of fuzzy rough sets is investigated.