Combination of Rough and Fuzzy Sets Based on α-Level Sets

A fuzzy set can be represented by a family of crisp sets using its α-level sets, whereas a rough set can be represented by three crisp sets. Based on such representations, this paper examines some fundamental issues involved in the combination of rough-set and fuzzy-set models. The rough-fuzzy-set and fuzzy-rough-set models are analyzed, with emphasis on their structures in terms of crisp sets. A rough fuzzy set is a pair of fuzzy sets resulting from the approximation of a fuzzy set in a crisp approximation space, and a fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a crisp set in a fuzzy approximation space. The approximation of a fuzzy set in a fuzzy approximation space leads to a more general framework. The results may be interpreted in three different ways

Logical and algebraic Techniques for Rough Set Data Analysis

this paper, we shall give an introduction to, and an overview of, the various relational, algebraic, and logical tools that are available to handle rule based reasoning