Boundary region based rough sets

Rough sets are mainly defined by means of neighborhood systems. In this paper, a type of novel definition of the approximation operators is proposed via a predefined boundary region based on a binary relation. Then, the comparison between the new definitions and the original ones is made. The necessary and sufficient conditions of their equipollence are obtained. This paper also gives the definitions of boundaries based on covering. By employing the boundaries, a type of novel definition of the approximation operators based on covering is proposed. It is shown that the novel operators are equivalent to a type of covering approximation operators. Finally, with the aid of the boundary region, the relation between the general binary relations and the covering-based approximation operators is examined.