The objective of this chapter is to provide a semantic framework for fuzzy sets in the theory of rough sets. Rough membership functions are viewed as a special type of fuzzy membership functions interpretable using conditional probabilities. The relationships between fuzzy membership functions and rough membership functions, between core and support of fuzzy set theory and lower and upper appro...

Fuzzy sets and rough sets are known as uncertainty models. They are proposed to treat different aspects of uncertainty. Therefore, it is natural to combine them to build more powerful mathematical tools for treating problems under uncertainty. In this chapter, we describe the state of the art in the combinations of fuzzy and rough sets dividing into three parts. In the first part, we first desc...

Fuzzy covering-based rough set models are hybrid models using both rough set and fuzzy set theory. The former is often used to deal with uncertain and incomplete information, while the latter is used to describe vague concepts. The study of fuzzy rough set models has provided very good tools for machine learning algorithms such as feature and instance selection. In this article, we discuss diff...

Fuzzy-rough sets play an important role in dealing with imprecision and uncertainty for discrete and real-valued or noisy data. However, there are some problems associated with the approach from both theoretical and practical viewpoints. These problems have motivated the hybridisation of fuzzy-rough sets with kernel methods. Existing work which hybridises fuzzy-rough sets and kernel methods emp...

In order to involve user knowledge in determining equality of sets, which may not be equal in the mathematical sense, three types of approximate (rough) equalities were introduced by Novotny and Pawlak ([8, 9, 10]). These notions were generalized by Tripathy, Mitra and Ojha ([13]), who introduced the concepts of approximate (rough) equivalences of sets. Rough equivalences capture equality of se...

The optimistic multigranulation T-fuzzy rough set model was established based on multiple granulations under T-fuzzy approximation space by Xu et al., 2012. From the reference, a natural idea is to consider pessimistic multigranulation model in T-fuzzy approximation space. So, in this paper, the main objective is to make further studies according to Xu et al., 2012. The optimistic multigranulat...

This paper considers the relations among L -fuzzy sets, rough sets and hyperring theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy hyperideals of a hyperring is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy hyperideal are provided and some significant propert...

Knowledge about a system contains ambiguity and vagueness. Fuzzy ontologies using rough concepts are able to deal with imprecise and vague knowledge. Several important amount of work has been carried out in the field of fuzzy ontologies during the last years and they have been successfully used in several applications, but appropriate methodology to represent fuzzy rough concepts using annotati...

This paper reviews and compares theories of fuzzy sets and rough sets. Two approaches for the formulation of fuzzy sets are reviewed, one is based on many-valued logic and the other is based on modal logic. Two views of rough sets are presented, set-oriented view and operator-oriented view. Rough sets under set-oriented view are closely related to fuzzy sets, which leads to non-truth-functional...

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough sets and rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a class of revised...