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In Pawlak rough sets, the structure of the definable set families is simpleand clear, but in generalizing rough sets, the structure of the definable setfamilies is a bit more complex. There has been much research work focusing onthis topic. However, as a fundamental issue in relation based rough sets, underwhat condition two relations induce the same definable set family has not beendiscussed. In this paper, based on the concept of the closure of relations, wepresent a necessary and sufficient condition for two relations to induce thesame definable set family.

A necessary and sufficient condition for two relations to induce the same definable set family