Partial Approximative Set Theory: A Generalization of the Rough Set Theory

Abstract

The paper presents a generalization of the classical rough set theory, called the partial approximative set theory (PAST). According to Pawlak’s rough set theory, the vagueness of a subset of a finite universe U is defined by the difference of its upper and lower approximations with respect to a σ-algebra generated by an equivalence relation on U. There are two most natural ways of the generalization of this idea. In particular, the equivalence relation is replaced by either any other type of binary relations on U or an arbitrary covering of U. In this paper, our starting point will be a partial covering of an arbitrary universe. In general, the family of sets neither covers the universe nor forms a σ-algebra. We will put our discussions into an overall treatment called the general set theoretic approximation framework. We will investigate under what conditions our generalized upper and lower approximation pair forms Galois connection.