New Generalized Definitions of Rough Membership Relations and Functions from Topological Point of View

In this paper, we shall integrate some ideas in terms of concepts in topology. In fact, we introduce two different views to define generalized membership relations and functions as mathematical tools to classify the sets and help for measuring exactness and roughness of sets. Moreover, we define several types of fuzzy sets. Comparisons between the induced operations were discussed. Finally, many results, examples and counter examples to indicate connections are investigated

On generalizing covering approximation space

In this paper, we present the covering rough sets based on neighborhoods by approximation operations as a new type of extended covering rough set models. In fact, we have introduced generalizations to W. Zhu approaches (Zhu, 2007). Based on the notion of neighborhood induced from any binary relation, four different pairs of dual approximation operators are defined with their properties being discussed. The relationships among these operators are investigated. Finally, an interesting theorem to generate different topologies is provided. Comparisons between these topologies are discussed. In addition, several examples and counter examples to indicate counter connections are investigated

Rough set theory for topological spaces

AbstractThe topology induced by binary relations is used to generalize the basic rough set concepts. The suggested topological structure opens up the way for applying rich amount of topological facts and methods in the process of granular computing, in particular, the notion of topological membership functions is introduced that integrates the concept of rough and fuzzy sets

Generalized covering approximation space and near concepts with some applications

In this paper, we shall integrate some ideas in terms of concepts in topology. First, we introduce some new concepts of rough membership relations and functions in the generalized covering approximation space. Second, we introduce some topological applications namely “near concepts” in the generalized covering approximation space. Accordingly, several types of fuzzy sets are constructed. The basic notions of near approximations are introduced and sufficiently illustrated. Near concepts are provided to be easy tools to classify the sets and to help for measuring exactness and roughness of sets. Many proved results, examples and counter examples are provided. Finally, we give two practical examples to illustrate our approaches