The Extension Of The Vague Set And Its Rough Approximation Theory

Vague sets, as extension of the fuzzy sets, since it could provide the informationof the membership degree, the non-membership degree and the hesitation degree, be-comes more and more flexible and efective in processing incomplete and uncertaininformation data. However, considering that vague sets’ representation is alwaysrestricted by the policy makers’ historical knowledge, perceptual judgement andother factors in the game playing, benefit groups’ voting or decision making pro-cess, we improve the relationship between the membership and the non-membershipdegree for the vague sets, and define the generalized vague sets by means of fuzzylogic non-portal operators in the paper. Moreover, the generalized vague rough setsystem and the rough generalized vague set system are discussed using both the con-structive and axiomatic approaches, respectively. In the constructive approach, theconcepts of the generalized vague rough sets and the rough generalized vague setsare defined and the properties are examined. In the axiomatic approach, the equiva-lent characterizations between the diferent axiomatic approximation operators andthe corresponding fuzzy relations and crisp binary relations are investigated by thegeneralized vague rough set algebras and the rough generalized vague set algebrasproposed in this paper, where the fuzzy relations and the crisp binary relations areinduced by the corresponding axiomatic operators. Meanwhile, the roughness mea-sure methods of the generalized vague sets are studied. Finally, a numerical exampleis given.