Covering soft rough sets and its topological properties with application

Soft rough fuzzy sets and soft fuzzy rough sets

The concept of soft set was presented as a new mathematical technique for dealing with uncertainty. Further soft set was developed and extended in the topological space to resolve certain complicated problems. We present new concepts such as soft ε-open set, soft εI - open set, soft εs-open set, and soft εsI-open set in this paper. The properties of certain known soft sets, as well as their relationships to them, are investigated. We also describe and analyze soft ε-pu continuous functions as well as their fundamental properties.

In [8], Császár introduced some generalized open sets in generalized topological spaces, namely, μ-semi-open, μ-preopen, μ-β-open, and μ-α-open. The primary purpose of this paper is to study some new properties and characterizations of these concepts as we also introduce and study a new generalized open set in generalized topological spaces, namely, μ-b-open. We also investigate some relationships between these concepts.

Emerging trends in soft set theory and related topics.

Fuzzy set theory, rough set theory and soft set theory are all generic mathematical tools for dealing with uncertainties. There has been some progress concerning practical applications of these theories, especially, the use of these theories in decision making problems. In the present article, we review some decision making methods based on (fuzzy) soft sets, rough soft sets and soft rough sets. In particular, we provide several novel algorithms in decision making problems by combining these kinds of hybrid models. It may be served as a foundation for developing more complicated soft set models in decision making.

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

Soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. The main purpose of soft rough set is reducing the soft boundary region by increasing the lower approximation and decreasing the upper approximation. This paper is devoted to the further generalization of the soft rough set theory by using the ideal notion to reduce the soft boundary region. A new soft rough set model, called soft ideal approximation space, is proposed and its properties are derived. The new soft ideal lower and upper approximation operators are presented and their related properties are surveyed.

The significance fore supra topological spaces as a subject of study cannot be overstated, as they represent a broader framework than traditional topological spaces. Numerous scholars have proposed extension to supra open sets, including supra semi open sets, supra per open and others. In this research, a notion for ⱨ-supra open created within the generalizations of the supra topology of sets. Our investigation involves harnessing this style of sets to introduce modern notions in these spaces, specifically supra ⱨ - interior, supra ⱨ - closure, supra ⱨ - limit points, supra ⱨ - boundary points and supra ⱨ - exterior of sets. It has been examining the relationship with supra open. The research was also enriched with many of characteristics of each concept. Building upon this set classification, we introduced several kinds of maps like supra ⱨ - continuous, supra ⱨ - open, supra ⱨ - tentative, supra ⱨ -globally and supra ⱨ - homeomorphism. Additionally, we have proven a collection of useful relationships for the aforementioned of functions. Furthermore, the research was enhanced with illustrative and refuting examples.

Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems.

In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

In this paper, we introduced study about a basic properties of adherent points by the class B*c – open. Also, we showed properties accumulation points by using B*c – open sets and we provided the relation among adherent points, β – adherent points and B*c – adherent points. Also, accumulation points, β – accumulation points and B*c – accumulation points.

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.

In the current study, the idea of S ψ φ -open sets is used to introduce some new types of nearly open sets in topological space, where ψ = { θ , δ , τ } and φ = { θ , δ , τ , α , a } , thereby illustrating some of their basic qualities and the links between them. Some characterizations, topological properties, mappings, and separation axioms of these new concepts are also presented.

This paper investigates the diversified multi-attribute group decision-making problem which means that the different decision-makers have different evaluation attribute sets for all candidate alternatives. We analyze the diversified multi-attribute group decision-making problem from the perspective of granular computing models and creatively combine soft sets with multigranulation fuzzy rough sets to construct the multigranulation soft fuzzy rough set model. Subsequently, we define the optimistic upper and lower approximations and pessimistic upper and lower approximations with respect to multigranulation soft fuzzy rough set model. Meanwhile, we discuss some important mathematical conclusions and properties based on the established model. Then, we propose a novel approach to the diversified multi-attribute group decision-making problem based on multigranulation soft fuzzy rough set and TODIM method, which fully consider the reference dependence and loss aversion of decision-makers in the decision-making process. Finally, we use a numerical example of evaluation for the policy selection group decision-making problem to describe and explain the calculation principle and process and compare the results with existing related work, and our approach has great stability and rationality. And the contributions of this paper include the following two points: (1) construct the multigranulation soft fuzzy rough set model; (2) combine the established model with TODIM method to provide a novel perspective to solve the diversified multi-attribute group decision-making problem.

Soft binary relation is used to define new classes of soft sets, namely BR-soft simply open set and BR-soft simply* alpha open set, in topological rough approximation space over two different universes. The defined set provides information on the missing elements of a BR-soft set and can help in simplifying decision making. Approximation operators are defined and the characteristics of the proposed sets are studied with examples. The relationship between the defined sets and other soft sets is brought out. An accuracy check was done to compare the proposed method with other methods. It is identified that the proposed method is more accurate.