Fuzzy Approximation Operators Based on Coverings

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfy many interesting properties. Many types of generalized rough set models have been developed in the literature. This paper presents a general framework for the study of rough intuitionistic fuzzy sets in which intuitionistic fuzzy sets are approximated with respect to a crisp approximation space. A pair of lower and upper rough intuitionistic fuzzy approximation operators are first defined. Properties of rough intuitionistic fuzzy approximations are then examined. The relationships between special types of crisp binary relations and properties of rough intuitionistic fuzzy approximation operators are further established.

On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

Typical hesitant fuzzy set is a kind of generalization of classical fuzzy set by possessing a membership degree, called as typical hesitant fuzzy element, of finite non-empty subset of the unitary interval. This paper studies the constructive approach to rough set approximation operators in typical hesitant fuzzy background. Firstly, two novel partial orders are introduced to compare two arbitrary typical hesitant fuzzy elements and typical hesitant fuzzy sets, meanwhile, the basic operations, including intersection, union and α−level sets of typical hesitant fuzzy sets are then proposed and their properties are studied in detail. Secondly, typical hesitant fuzzy rough sets are introduced and studied via the new operations of intersection and union. Furthermore, typical hesitant fuzzy rough approximation operators are represented by the rough approximation operators of the α−level set of typical hesitant fuzzy set. Finally, the connections between typical hesitant fuzzy (and crisp, respectively) relations and typical hesitant fuzzy rough (and rough typical hesitant fuzzy, respectively) approximation operators are further established.

On (IO,O)-fuzzy rough sets based on overlap functions

In rough set theory, the lower and upper approximation operators defined by binary relations satisfy many interesting properties. Various generalizations of Pawlak's rough approximations have been made in the literature over these years. This paper proposes a general framework for the study of intuitionistic fuzzy rough approximation operators based on intuitionistic fuzzy triangle norm. In the constructive approach, a pair of lower and upper induced from intuitionistic fuzzy relation are defined. Basic properties of intuitionsitic fuzzy rough approximation operators are then examined. By introducing intuitionistic fuzzy residual implication, Further properties of intuitionistic fuzzy rough approximation operators are then investigated. we propose that classical rough set and fuzzy rough are special types of intuitionistic fuzzy rough set based on intuitionistic fuzzy triangle norm.

A fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a fuzzy/crist set in a fuzzy approximation space. A fuzzy rough set algebra is a fuzzy set algebra with added dual pair of fuzzy rough approximation operators. In this paper, we study the mathematical structures of fuzzy rough set algebras in infinite universes of discourse. We first define the concept of fuzzy rough set algebras by the axiomatic approach. We then examine the properties of fuzzy rough approximation operators in different types of fuzzy rough set algebras. We also prove that if a system \(({\cal F}(U), \cap, \cup, \sim, L, H)\) is a (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra then the derived system \(({\cal F}(U), \cap, \cup, \sim, LL, HH)\) is also a (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra.KeywordsApproximation operatorsFuzzy rough setsFuzzy setsFuzzy rough set algebrasRough sets

A rough fuzzy set is the result of approximation of a fuzzy set with respect to a crisp approximation space. In this paper, we investigate topological structures of rough fuzzy sets. We first show that a reflexive crisp rough approximation space can induce a fuzzy Alexandrov space. We then prove that the lower and upper rough fuzzy approximation operators are, respectively, the fuzzy interior operator and fuzzy closure operator if and only if the binary relation in the crisp approximation space is reflexive and transitive. Finally, we examine that a similarity crisp approximation space can produce a fuzzy clopen topological space.KeywordsApproximation operatorsBinary relationsFuzzy topologiesRough fuzzy setsRough sets

Abstract: Machine learning can extract desired knowledge from training examples and ease the development bottleneck in building expert systems. Most learning approaches derive rules from complete and incomplete data sets. If attribute values are known as possibility distributions on the domain of the attributes, the system is called an incomplete fuzzy information system. Learning from incomplete fuzzy data sets is usually more difficult than learning from complete data sets and incomplete data sets. In this paper, we deal with the problem of producing a set of certain and possible rules from incomplete fuzzy data sets based on rough sets. The notions of lower and upper generalized fuzzy rough approximations are introduced. By using the fuzzy rough upper approximation operator, we transform each fuzzy subset of the domain of every attribute in an incomplete fuzzy information system into a fuzzy subset of the universe, from which fuzzy similarity neighbourhoods of objects in the system are derived. The fuzzy lower and upper approximations for any subset of the universe are then calculated and the knowledge hidden in the information system is unravelled and expressed in the form of decision rules.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

The axiomatic approach is more appropriate than constructive approach for studying the algebraic structure of rough sets. In this paper, the more simple axiomatic characterizations of (υ σ)-fuzzy rough approximation operators are explored where υ is a residuated implicator and σis its dual implicator. Firstly, we review the existing independent axiomatic sets to characterize various types of υ-lower and σ-upper fuzzy rough approximation operators. Secondly, we present one-axiom characterizations of (υ σ)-fuzzy rough approximation operators constructed by a serial fuzzy relation on two universes. Furthermore, we show that (υ σ)-fuzzy rough approximation operators, corresponding to reexive, symmetric and T-transitive fuzzy relations, can be presented by only two axioms respectively. We conclude the paper by introducing some potential applications and future works.

Hesitant fuzzy set is a generalization of the classical fuzzy set by returning a family of the membership degrees for each object in the universe. Since how to use the rough set model to solve fuzzy problems plays a crucial role in the development of the rough set theory, the fusion of hesitant fuzzy set and rough set is then firstly explored in this paper. Both constructive and axiomatic approaches are considered for this study. In constructive approach, the model of the hesitant fuzzy rough set is presented to approximate a hesitant fuzzy target through a hesitant fuzzy relation. In axiomatic approach, an operators-oriented characterization of the hesitant fuzzy rough set is presented, that is, hesitant fuzzy rough approximation operators are defined by axioms and then, different axiom sets of lower and upper hesitant fuzzy set-theoretic operators guarantee the existence of different types of hesitant fuzzy relations producing the same operators.

This paper presents a general framework for the study of fuzzy rough sets in which constructive approach is used. In the approach, a pair of lower and upper general fuzzy rough approximation operator in the lattice L is defined. Furthermore, the entire property and connection between the set of logical operator and the set of its corresponding general fuzzy rough approximation operator are examined, and we prove that they are 1-1 mapping. In addition, the structural theorem of negator operator is given. At last, the decomposition and synthesize theorem of general fuzzy rough approximation operator are proved. That for how to promote general rough approximation operator to suitable general fuzzy rough approximation operator, that is, how to select logical operator, provides theory foundation.

This paper aims to originate two new notions that are soft rough Pythagorean fuzzy set and Pythagorean fuzzy soft rough set, and investigate some important properties of soft rough Pythagorean fuzzy set and Pythagorean fuzzy soft rough set in detail. Furthermore, classical representations of Pythagorean fuzzy soft rough approximation operators are presented. Then the proposed operators are applied on decision-making problem in which the experts provide their preferences in Pythagorean fuzzy soft rough environment. Finally through an illustrative example, it is shown that how the proposed operators work in decision-making problems.

This paper studies linear structure of fuzzy rough sets. Using the composition of fuzzy relation and a fuzzy set in a given universal set, a new representation of the fuzzy rough set upper approximations is given. According to this view of linearity, a simplified structure is obtained for the fuzzy rough set lower and upper approximations.

Given a widespread interest in rough sets as being applied to various tasks of data analysis it is not surprising at all that we have witnessed a wave of further generalizations and algorithmic enhancements of this original concept. This paper proposes an interval-valued intuitionistic fuzzy rough model by means of integrating the classical Pawlak rough set theory with the interval-valued intuitionistic fuzzy set theory. Firstly, some concepts and properties of interval-valued intuitionistic fuzzy set and interval-valued intuitionistic fuzzy relation are introduced. Secondly, a pair of lower and upper interval-valued intuitionistic fuzzy rough approximation operators induced from an interval-valued intuitionistic fuzzy relation is defined, and some properties of approximation operators are investigated in detail. Furthermore, by introducing cut sets of interval-valued intuitionistic fuzzy sets, classical representations of interval-valued intuitionistic fuzzy rough approximation operators are presented. Finally, the connections between special interval-valued intuitionistic fuzzy relations and interval-valued intuitionistic fuzzy rough approximation operators are constructed, and the relationships of this model and the others rough set models are also examined.