Properties of Lower and Upper Approximation Operators under Various Kinds of Relations

Abstract

One of the direction of rough sets theory is to extend the equivalence relation, using more general binary relation such as the partial order relation, tolerance relation or similarity relations instead of the strict equivalence relation. So the scope of application of rough sets theory could be extended. But in the use of these more general relation instead of equivalence relation, some good properties of the original Pawlak approximation space �� , �� may no longer be satisfied. In this paper, various properties of two pairs of lower and upper approximation operators and the relationship between them are acquired through analysis and proof. The two pairs of lower and upper approximation operator will have good properties when the binary relation R only satisfies reflexivity, and the two pairs of approximation operators are the same under the equivalence relation. Theoretical analysis shows that, in the process of extending classical rough set theory to generalized rough set theory, reflexivity is a minimum conditions to be satisfied, under this condition, the lower and upper approximation operators meeting the corresponding properties can be chosen to adapt to the requirement of practical application.