Development of Near Sets Within the Framework of Axiomatic Fuzzy Sets

Near sets result from a generalization of rough sets, which introduced by Peters in 2006, and later formally defined in 2007. Near set theory provides a new framework for representation of objects characterized by the features that describe them. AFS (Axiomatic Fuzzy Set) theory was proposed by Liu (1998), which is a semantic methodology relating to the fuzzy theory. In this paper, a new version of near sets based on AFS theory is established, in which every object has an AFS fuzzy description with definitely semantics. The proposed approach to assessing the nearness (closeness) of objects is not defined directly using a distance metric, but depend on similarity of their fuzzy descriptions. It is also a natural linguistic description that is similar to humans perception. Moreover, an approach to set approximation based on the union of families of objects with similar fuzzy descriptions is given. The near sets based on AFS theory can be viewed as a new development of near sets within the fuzzy context.