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 setmodels concern rough approximations based onmultiple 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 multigranulation rough set approximations are investigated due to the fuzzy binary relations in aMGFAS. 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.