Rough approximations of vague sets in fuzzy approximation space
The combination of the rough set theory, vague set theory and fuzzy set theory is a novel research direction in dealing with incomplete and imprecise information. This paper mainly concerns the problem of how to construct rough approximations of a vague set in fuzzy approximation space. Firstly, the beta-operator and its complement operator are introduced, and some new properties are examined. Secondly, the approximation operators are constructed based on beta-(complement) operator. Meantime, lambda-lower (upper) approximation is firstly proposed, and then some properties of two types of approximation operators are studied. Afterwards, for two different kinds of approximation operators, we introduce two roughness measure methods of the same vague set and discuss a property. Finally, an example is given to illustrate how to calculate the rough approximations and roughness measure of a vague set using the beta-(complement) product between two fuzzy matrixes. The results show that the proposed rough approximations and roughness measure of a vague set in fuzzy environment are reasonable. Crown Copyright (C) 2010 Published by Elsevier Inc. All rights reserved.
rough approximation vague sets fuzzy approximation space beta-(complement) operator roughness measure fuzzy equivalence relation