On generalization of rough sets by using two different methods

Rough set theory was introduced by Pawlak in 1982 to handle imprecision, vagueness, and uncertainty in data analysis. It is dealing with vagueness (ambiguous) of the set by using the concept of the lower and upper approximations of objects based on an equivalence relation. The main idea of rough sets corresponds to study these approximations. So, in this paper, we generalize these approximations in the frameworks of topological spaces. The lower and upper approximations of Pawlak's model are replaced by interior and closure notions of the topological space. The set approximations are defined using the new topological notions delta(beta j)-open sets and boolean AND(beta j)-sets. Such techniques open the way for more topological applications in rough context and help in formalizing many applications from real-life data. The current extension approximations are satisfied all properties of original rough set theory without any conditions or restrictions. Comparisons between the current approximations and the previous one are introduced and shown to be more general.