Rough Set Approximations In Multi-Granulation Fuzzy Approximation Spaces

Pawlak's rough set model considers the rough approximations based on an equivalence relation. Multi-granulation rough set models concern rough approximations based on multiple equivalence relations. In this paper, we examine six types of rough set approximations in multi-granulation fuzzy approximation spaces (MGFASs). We construct a partition of the given universe based on a fuzzy binary relation in a fuzzy approximation space. Based on the partition, we introduce a pair of rough set approximations. In a multi-granulation fuzzy approximation space, by a family of fuzzy binary relations, we introduce two kinds of rough set approximations in terms of the union and intersection of fuzzy relations, respectively. A pair of rough set approximations based on the family of fuzzy binary relations is also discussed. Furthermore, the optimistic and pessimistic multi-granulation rough set approximations are investigated due to the fuzzy binary relations in a MGFAS. Properties of these rough set approximations are demonstrated. Finally, we examine relationships of them. It is proved that the lower and upper approximations generated by a family of fuzzy binary relations are the pair nearest to the undefinable set, and the pessimistic multi-granulation lower and upper approximations are the pair farthest to the undefinable set.