Open accessJournal Article

# New Rough Approximations Based on E-Neighborhoods

02 Mar 2021-Complexity (Hindawi Limited)-Vol. 2021, pp 1-6
Abstract: This paper puts forward some rough approximations which are motivated from topology. Given a subset , we can use 8 types of - neighborhoods to construct approximations of an arbitrary on the one hand. On the other hand, we can also construct approximations relying on a topology which is induced by an - neighborhood. Properties of these approximations and relationships between them are studied. For convenience of use, we also give some useful and easy-to-understand examples and make a comparison between our approximations and those in the published literature.

Topics: Topology (chemistry) (55%)
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Journal Article
Tareq M. Al-shami1Institutions (1)
Abstract: The rough set theory is a nonstatistical mathematical approach to address the issues of vagueness and uncertain knowledge The rationale of this theory relies on associating a subset with two crisp sets called lower and upper approximations which are utilized to determine the boundary region and accuracy measure of that subset Neighborhoods systems are pivotal technique to reduce the boundary region and improve the accuracy measure Therefore, we aim through this paper to introduce new types of neighborhoods called containment neighborhoods (briefly, Cj-neighborhoods) They are defined depending on the inclusion relations between j-neighborhoods under arbitrary binary relation We study their relationships with some previous types of neighborhoods, and determine the conditions under which they are equivalent Then, we applied Cj-neighborhoods to present the concepts of Cj-lower and Cj-upper approximations and reveal main properties with the help of examples We also prove that a Cj-accuracy measure is the highest in cases of j=i,〈i〉 Furthermore, we compare our approach with two approaches given in published literatures and show that accuracy measure induced from our technique is the best Finally, we successfully applied Cj-neighborhoods, Nj-neighborhoods and Ej-neighborhoods in a medical application aiming to classify medical staff in terms of suspected infection with the new corona-virus (COVID-19)

Topics: Rough set (55%), , Binary relation (53%)

12 Citations

Journal Article
Abstract: Approximation space can be said to play a critical role in the accuracy of the set’s approximations. The idea of “approximation space” was introduced by Pawlak in 1982 as a core to describe informa...

Topics: Rough set (61%), Topology (chemistry) (53%), Core (graph theory) (53%) ... read more

4 Citations

Journal Article
Abstract: In this paper, with the aid of fuzzy soft $$\beta$$ -neighborhoods, we introduce fuzzy soft covering-based multi-granulation fuzzy rough set models. We examine some of the relevant properties of fuzzy soft covering based on optimistic, pessimistic, and variable precision multi-granulation fuzzy rough set models. Then, we give fuzzy soft coverings based on $$\psi$$ -optimistic and $${\mathscr {D}}$$ -optimistic ( $$\psi$$ -pessimistic and $${\mathscr {D}}$$ -pessimistic) multi-granulation fuzzy rough sets from fuzzy soft measures. It also discusses the interactions between these forms of fuzzy soft coverings based on multi-granulation fuzzy rough sets. Eventually, we apply the proposed models for solving MAGDM problems. The effectiveness and feasibility of our approach are noted from the introduced comparisons between our method and some methods given in the previous studies.

Topics: Fuzzy logic (59%)

3 Citations

Open accessJournal Article
Abstract: The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of - adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.

Topics: Rough set (56%), Reflexive relation (53%), Binary relation (53%)

2 Citations

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Journal Article
Abstract: This paper proposes new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from the concept of ambiguity introduced in this paper. The new definitions are compared to the classical definitions and are shown to be more general, in the sense that they are the only ones which can be used for any type of indiscernibility or similarity relation.

Topics: , Rough set (53%)

907 Citations

Journal Article
Andrzej Skowron1, Jarosław Stepaniuk2Institutions (2)
Abstract: We generalize the notion of an approximation space introduced in [8] In tolerance approximation spaces we define the lower and upper set approximations We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems The tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process We obtain the solutions of the reduction problems by applying boolean reasoning [1] The solutions are represented by tolerance reducts and relative tolerance reducts

905 Citations

Open accessJournal Article
Yiyu Yao1Institutions (1)
Abstract: This paper presents a framework for the formulation, interpretation, and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations. A special class of neighborhood systems, called 1-neighborhood systems, is introduced. Three extensions of Pawlak approximation operators are analyzed. Properties of neighborhood and approximation operators are studied, and their connections are examined.

Topics: Operator theory (61%), Rough set (58%), Binary relation (51%)

865 Citations

Open accessJournal Article
Yiyu Yao1Institutions (1)
Abstract: This paper presents and compares two views of the theory of rough sets. The operator-oriented view interprets rough set theory as an extension of set theory with two additional unary operators. Under such a view, lower and upper approximations are related to the interior and closure operators in topological spaces, the necessity and possibility operators in modal logic, and lower and upper approximations in interval structures. The set-oriented view focuses on the interpretation and characterization of members of rough sets. Iwinski type rough sets are formed by pairs of definable (composed) sets, which are related to the notion of interval sets. Pawlak type rough sets are defined based on equivalence classes of an equivalence relation on the power set. The relation is defined by the lower and upper approximations. In both cases, rough sets may be interpreted by, or related to, families of subsets of the universe, i.e., elements of a rough set are subsets of the universe. Alternatively, rough sets may be interpreted using elements of the universe based on the notion of rough membership functions. Both operator-oriented and set-oriented views are useful in the understanding and application of the theory of rough sets.

Topics: , Rough set (68%), Family of sets (62%) ... read more

528 Citations

Journal Article
William Zhu1Institutions (1)
Abstract: Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. This paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering-based rough sets and the generalized rough sets based on binary relation. Firstly, we present basic concepts and properties of this kind of rough sets. Then we investigate the relationships between this type of generalized rough sets and other five types of covering-based rough sets. The major contribution in this paper is that we establish the equivalency between this type of covering-based rough sets and a type of binary relation based rough sets. Through existing results in binary relation based rough sets, we present axiomatic systems for this type of covering-based lower and upper approximation operations. In addition, we explore the relationships among several important concepts such as minimal description, reduction, representative covering, exact covering, and unary covering in covering-based rough sets. Investigation of this type of covering-based will benefit to our understanding of other types of rough sets based on covering and binary relation.