The Intuitionistic Fuzzy Rough Sets Based On K-step-relation

Rough set theory was originally proposed by Pawlak[1]. It is a new mathemat-ical approach to deal with incomplete and insufficient information. And it is also an extension of the classical set theory. This theory is currently emerging as a pow-erful tool with applications in the field of artificial intelligence such as automated knowledge acquisition, data mining, decision making and analysis. A key notion in Pawlak rough set model is equivalence relation. However, an equivalence rela-tion is so strigent that it may limit the application domain of the rough set model. For further extending the theories and application of rough sets, different kinds of generalization in rough set models were obtained. For example, we can combine intuitionistic fuzzy (IF) set theory and rough theory, then we obtain IF rough sets model.Fistly, a kind of intuitionistic fuzzy rough sets based onκ-step-relation is de-fined. Basic properties of the intuitionistic fuzzy rough approximation operators are then discussed in the case of reflexive, symmetric and transtitive relations. Further-more, We investigate the topological structures of intuitionistic fuzzy rough sets. And we see that an crisp relation can induce an IF topology on U. When the re-lation is symmetry, an IF open set is also closed set in (U,rRκ). And we see that an reflexive relation can also generate IF topological spaces and showes that the lower and upper IF rough approximation operators are, respectively, the interior and closure operatorsof the IF topological spaces.Secondly, the constructive and axiomatic approaches in the study of IF rough sets are discussed. In the constructive metheods, a pair of lower and upper approx-imation opetrators are defined using theκ-step fuzzy relation. Different classes of IF rough sets algebra are obtained from the different types of fuzzy relations. In the axiomatic approaches, a pair of dual generalized approximation operators which must be satisfied by the operators are given. The axioms guarantee the existence of certain types of relations that generate the same operators.Finally, a pair of lower and upper IF rough approximation operators induced from an arbitrary IF relation are defined. Some properties of the intuitionistic fuzzy rough approximation operators are then examined in the case of reflexive, symmetric and transtitive relations. And we see that an IF relation can induce an IF topology on U. In the finite space an reflexive relation can also generate IF topological spaces and the lower and upper IF rough approximation operators are, respectively, the interior and closure operators of the IF topological spaces.