Abstract
The generalized rough set theory based on binary relations is the improvement of Pawlak's rough set theory to deal with more complex practical problems which the latter one can not easily handle. For a given generalized approximation space, there are two traditional forms of definition of the lower approximation, it is important to reveal the inner relationship between the two definition forms. On the other hand, in many cases, people have to make decisions in practical problems based on binary relations. Therefore, it is also necessary to generalize a decision system in Pawlak' rough set theory. In this paper, the concept of a relation decision system is firstly introduced by generalizing a decision system in Pawlak' rough set theory. By the revisions of the traditional concepts, this paper then presents the essential relationship between the two forms of definition of the lower approximation and obtains some useful properties for a relation decision system
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