Generalization of rough fuzzy sets based on a fuzzy ideal

Journal of the Egyptian Mathematical Society

In this paper, we defined the fuzzy upper, fuzzy lower, and fuzzy boundary sets of a rough fuzzy set λ in a fuzzy approximation space (X, R). Based on λ and R, we introduced the fuzzy ideal approximation interior operator intlambda R and the fuzzy ideal approximation closure operator cl R λ. We joined the fuzzy ideal notion with the fuzzy approximation spaces, and then introduced the fuzzy ideal approximation closure and interior operators associated to a rough fuzzy set λ. Fuzzy ideal approximation connectedness and the fuzzy ideal approximation continuity between fuzzy ideal approximation spaces are introduced.

IEEE Transactions on Fuzzy Systems, 2005

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations,-similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.

Information Sciences, 2003

This paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized approximation operators is defined. The connections between fuzzy relations and fuzzy rough approximation operators are examined. In axiomatic approach, various classes of fuzzy rough approximation operators are characterized by different sets of axioms. Axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators.

International Journal of Fuzzy Logic System, 2021

Rough set theory and its hybrid models are new emerging techniques to deal with uncertainties, impreciseness and ambiguities. In past few years, researchers are taking keen interest in rough set theory, which is an important area of meteorological field especially in weather forecasting and in artificial intelligence. In the present communication basic concepts and some aspects of rough set theory are defined and explained. Its hybrid models namely fuzzy rough and rough fuzzy sets are described in detail. Fuzzy rough information measure is characterized and its application is studied with illustration. Approximate equalities using rough fuzzy and rough intuitionistic fuzzy sets are also given with the conclusion in the end.

Fundamenta Informaticae, 2015

The title of this note announces an attempt at pointing to particular points where rough set and fuzzy set approaches to decision making are close to each other, the closeness meaning that formal approaches to behaviour of the two at those points can be given in an analogous form. This by no means implies that the two can be unified as theories dealing with uncertainty. As the notion of truth for rough set decision rules is well established, we propose a notion of truth for fuzzy decision rules and we seek an analogy between the two. In order to introduce an analogous form of graded notion of truth for decision rules in both theories, we introduce a new context in which to set this notion. This context is based on our earlier results concerning rough mereological granular logics and their relevance for rough decision rules. To make our exposition satisfactorily complete, we recall our approach to granularity based on rough mereology.

Fuzzy Sets and Systems, 2014

The roughness of a set (according to the notion introduced by Pawlak in 1991) can be regarded as the MZ-distance between its upper and the lower approximations. With this idea in mind, we have generalized Pawlak's definition, by replacing the MZ-distance by a general "distance" measure. We also generalize the notion of roughness of fuzzy sets introduced by Huyhn and Nakamori in 2005.

Information Sciences, 1996

An integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set. Several properties of this new measure are established. Some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned.

Journal of Intelligent & Fuzzy Systems

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

Dutse Journal of Pure and Applied Sciences

Fuzzy T-rough set consists of a set ???? and a T-similarity relation on ????, where T is a lower semicontinuous triangular norm. In this paper, axiomatic definition for fuzzy ????-rough sets and its upper approximation operator were proposed. The method employed was by relaxing the arbitrary T and adopting its special case ???????? (product triangular norm). The results obtained suggests an easier way of being specific to the product case of fuzzy rough sets and computations regarding its upper approximation operators. Some important propositions and examples were also provided.

Information Sciences, 2005

Recently, an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets . Essentially, Banerjee and Pal's roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set. In this paper we first remark that this measure of roughness has several undesirable properties, and then propose a parameter-free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set. Several interesting properties of this new measure are examined. Furthermore, we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification.