Probabilistic rough sets characterized by vague sets

Rough set and vague set theories are used to study the different kinds of uncertainty in the information systems. They are generalizations of classical set theory for describing vagueness and uncertainty. The focus of this paper is to characterize probabilistic rough sets by vague sets. Firstly, the membership functions of vague sets are shown by conditional probabilities. Secondly, the concepts of variable precision lower and upper approximations of a probabilistic rough set are given by the lower and upper bounds of truth-membership and false-membership functions of a vague set. At last, the fuzziness in a probabilistic rough set can be induced by fuzzy entropies of a vague set.