160 Proceedings of National Conference on Mathematical & Computational Sciences- 6 th &7 th July, 2012 Adikavi Nannaya University, Rajahmundry [AP], India Two Way Naïve Intuitionistic Rough Fuzzy Groups B.Krishnaveni* 1 G.Ganesan* 2 * 1,2 Department of Mathematics, Adikavi Nannaya University, Rajahmundry, Andhra Pradesh ABSTRACT In 1982, Z Pawlak introduced the concept of rough sets. This theory found applications in approximating the essentials in given information system with respect to the available knowledge. In 1986, Attanasov introduced intuitionistic fuzzy sets by considering the membership grades of L. Zadeh along with non-membership grades. In 1989, Dubois and Prade introduced rough fuzzy sets by hybridizing fuzzy concepts and rough sets. In 2005,G.Ganesan et.al., dealt the group structure in the rough fuzzy sets and named them as rough fuzzy groups. Later the approach of algebraic structure has been extended by G.Ganesan et. al., into intuitionistic rough fuzzy group which brings out the algebraic aspects on intuitionistic rough fuzzy sets. In 2008, G.Ganesan et.al., contributed four way generalizations on intuitionistic rough fuzziness. In this paper, we apply the earlier algebraic aspects on the Two way generalizations on intuitionistic rough fuzzy sets. Keywords : Rough sets, rough fuzzy sets, rough fuzzy group, intuitionistic fuzzy sets, intuitionistic rough fuzzy sets 1. INTRODUCTION In 1982, Pawlak introduced the theory of rough sets [8,9]. This theory involves in several technical aspects related to decision making. The concept of rough sets has been viewed algebraically as well as topologically by the mathematicians. Parallel to this theory, L.Zadeh’s Fuzzy concepts have been noticed as an efficient tool where the Conventional Boolean concepts fail. Hence considering the importance of fuzziness in rough computing, in 1989,Dubios and Prade hybridized both approaches and introduced rough-fuzziness and fuzzy-roughness. In 2005, G.Ganesan et.al., introduced a Naïve group structure in rough fuzzy set and named it as rough fuzzy group. Since Zadeh’s concepts found to be firm in defining non-membership grades, to allow them to be flexible, in 1985, Attanasov introduced Intuitionistic fuzziness. In 2008, G.Ganesan et. al., discussed the basic definitions of Dubois and Prade into four ways and contributed four way intuitionistic rough fuzziness. In this paper, the work of Rough Fuzzy Groups is extended into Two way Naïve Intuitionistic rough fuzzy groups. This paper comprises of six sections. Section two deals with the basics of fuzzy sets and intuitionistic fuzzy sets; Section three deals with rough sets, rough fuzzy sets, and intuitionistic rough fuzzy sets. Section four deals with rough fuzzy group and its generalizations; Section Five deals with Two Way Naïve Generalized Intuitionistic Rough Fuzzy Group structure. 2. FUZZY SETS AND INTUITIONISTIC FUZZY SETS In this section, we describe the basic concepts of fuzzy sets and intuitionistic fuzzy sets. 2.1Fuzzy Sets In 1965, Zadeh [7] introduced fuzzy sets by generalizing Classical sets. For a given universe of discourse U, a classical subset A of U is expressed in termed of its characteristic function A  defined by       A x if A x if x A 0 1 ) (  This representation lacks in specifying the significance of an element in the set which made Zadeh to work on grades of memberships for the elements of a set. These membership grades allow values ranging from