scholarly journals Categorical Properties of Soft Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Min Zhou ◽  
Shenggang Li ◽  
Muhammad Akram
Keyword(s):  
Set Theory ◽  
Soft Set ◽  
Soft Sets ◽  
Categorical Theory ◽  
Categorical Properties ◽  
Topological Construct ◽  
Categorical Framework ◽  
Set Relations ◽  
Cartesian Closed ◽  
Injective Hulls

The present study investigates some novel categorical properties of soft sets. By combining categorical theory with soft set theory, a categorical framework of soft set theory is established. It is proved that the categorySFunof soft sets and soft functions has equalizers, finite products, pullbacks, and exponential properties. It is worth mentioning that we find thatSFunis both a topological construct and Cartesian closed. The categorySRelof soft sets andZ-soft set relations is also characterized, which shows the existence of the zero objects, biproducts, additive identities, injective objects, projective objects, injective hulls, and projective covers. Finally, by constructing proper adjoint situations, some intrinsic connections betweenSFunandSRelare established.

On soft set-valued maps and integral inclusions

2021 ◽  
pp. 1-15
Author(s):  
Monairah Alansari ◽  
Shehu Shagari Mohammed ◽  
Akbar Azam
Keyword(s):  
Fixed Point ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Fixed Point Theorems ◽  
Soft Set ◽  
Mathematical Tool ◽  
Multivalued Maps ◽  
Soft Sets ◽  
Special Cases ◽  
New Development

As an improvement of fuzzy set theory, the notion of soft set was initiated as a general mathematical tool for handling phenomena with nonstatistical uncertainties. Recently, a novel idea of set-valued maps whose range set lies in a family of soft sets was inaugurated as a significant refinement of fuzzy mappings and classical multifunctions as well as their corresponding fixed point theorems. Following this new development, in this paper, the concepts of e-continuity and E-continuity of soft set-valued maps and αe-admissibility for a pair of such maps are introduced. Thereafter, we present some generalized quasi-contractions and prove the existence of e-soft fixed points of a pair of the newly defined non-crisp multivalued maps. The hypotheses and usability of these results are supported by nontrivial examples and applications to a system of integral inclusions. The established concepts herein complement several fixed point theorems in the framework of point-to-set-valued maps in the comparable literature. A few of these special cases of our results are highlighted and discussed.

scholarly journals An algorithm for fuzzy soft set based decision making approach

2020 ◽  
Vol 30 (1) ◽  
pp. 59-70
Author(s):  
Shehu Mohammed ◽  
Akbar Azam
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Group Decision Making ◽  
Group Decision ◽  
Soft Set ◽  
Mathematical Tool ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Fuzzy Soft Sets ◽  
Priority Table

The notion of soft set theory was initiated as a general mathematical tool for handling ambiguities. Decision making is viewed as a cognitive-based human activity for selecting the best alternative. In the present time, decision making techniques based on fuzzy soft sets have gained enormous attentions. On this development, this paper proposes a new algorithm for decision making in fuzzy soft set environment by hybridizing some existing techniques. The first novelty is the idea of absolute scores. The second concerns the concept of priority table in group decision making problems. The advantages of our approach herein are stronger power of objects discrimination and a well-determined inference.

scholarly journals A note on soft near-rings and idealistic soft near-rings

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 53-68 ◽  
Author(s):  
Aslıhan Sezgin ◽  
Osman Atagün ◽  
Emin Aygün
Keyword(s):  
Set Theory ◽  
Theoretical Aspect ◽  
Soft Set ◽  
Mathematical Tool ◽  
Soft Sets ◽  
Theoretical Approaches

Molodtsov introduced the theory of soft sets, which can be seen as an effective mathematical tool to deal with uncertainties, since it is free from the difficulties that the usual theoretical approaches have troubled. In this paper, we apply the definitions proposed by Ali et al. [M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009), 1547-1553] to the concept of soft near- rings and substructures of soft near-rings, proposed by Atag?n and Sezgin [A. O. Atag?n and A. Sezgin, Soft Near-rings, submitted] and show them with illustrating examples. Moreover, we investigate the properties of idealistic soft near-rings with respect to the near-ring mappings and we show that the structure is preserved under the near-ring epimorphisms. Main purpose of this paper is to extend the study of soft near-rings from a theoretical aspect.

scholarly journals On Fuzzy Soft Sets

2009 ◽  
Vol 2009 ◽  
pp. 1-6 ◽  
Author(s):  
B. Ahmad ◽  
Athar Kharal
Keyword(s):  
Set Theory ◽  
Soft Set ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Fuzzy Soft Sets

We further contribute to the properties of fuzzy soft sets as defined and studied in the work of Maji et al. ( 2001), Roy and Maji (2007), and Yang et al. (2007) and support them with examples and counterexamples. We improve Proposition 3.3 by Maji et al., (2001). Finally we define arbitrary fuzzy soft union and fuzzy soft intersection and prove DeMorgan Inclusions and DeMorgan Laws in Fuzzy Soft Set Theory.

scholarly journals A new approach to bipolar soft sets and its applications

2015 ◽  
Vol 07 (04) ◽  
pp. 1550054 ◽  
Author(s):  
Faruk Karaaslan ◽  
Serkan Karataş
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Soft Set ◽  
Mathematical Tool ◽  
Soft Sets ◽  
New Approach ◽  
Basic Properties ◽  
Set Operations ◽  
First Results

Molodtsov [Soft set theory-first results, Comput. Math. App. 37 (1999) 19–31] proposed the concept of soft set theory in 1999, which can be used as a mathematical tool for dealing with problems that contain uncertainty. Shabir and Naz [On bipolar soft sets, preprint (2013), arXiv:1303.1344v1 [math.LO]] defined notion of bipolar soft set in 2013. In this paper, we redefine concept of bipolar soft set and bipolar soft set operations as more functional than Shabir and Naz’s definition and operations. Also we study on their basic properties and we present a decision making method with application.

scholarly journals A Soft Interval Based Decision Making Method and Its Computer Application

2021 ◽  
Vol 46 (3) ◽  
pp. 273-296
Author(s):  
Gözde Yaylalı ◽  
Nazan Çakmak Polat ◽  
Bekir Tanay
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Soft Set ◽  
Computer Application ◽  
Complex Data ◽  
Soft Sets ◽  
Mathematical Algorithm ◽  
Huge Data ◽  
Special Cases ◽  
General Method

Abstract In today’s society, decision making is becoming more important and complicated with increasing and complex data. Decision making by using soft set theory, herein, we firstly report the comparison of soft intervals (SI) as the generalization of interval soft sets (ISS). The results showed that SIs are more effective and more general than the ISSs, for solving decision making problems due to allowing the ranking of parameters. Tabular form of SIs were used to construct a mathematical algorithm to make a decision for problems that involves uncertainties. Since these kinds of problems have huge data, constructing new and effective methods solving these problems and transforming them into the machine learning methods is very important. An important advance of our presented method is being a more general method than the Decision-Making methods based on special situations of soft set theory. The presented method in this study can be used for all of them, while the others can only work in special cases. The structures obtained from the results of soft intervals were subjected to test with examples. The designed algorithm was written in recently used functional programing language C# and applied to the problems that have been published in earlier studies. This is a pioneering study, where this type of mathematical algorithm was converted into a code and applied successfully.

scholarly journals Bipolar Soft Topological Spaces

2020 ◽  
Vol 13 (2) ◽  
pp. 227-245
Author(s):  
Asmaa Fadel ◽  
Syahida Che Dzul-Kifli
Keyword(s):  
Set Theory ◽  
Soft Set ◽  
The Other ◽  
Topological Spaces ◽  
Mathematical Tool ◽  
Soft Sets ◽  
Positive Information ◽  
Soft Limit ◽  
Soft Point ◽  
Soft Topological Space

Bipolar soft set theory is a mathematical tool associates between bipolarity and soft set theory, it is defined by two soft sets one of them gives us the positive information where the other gives us the negative. The goal of our paper is to define the bipolar soft topological space on a bipolar soft set and study its basic notions and properties. We also investigate the definitions of: bipolar soft interior, bipolar soft closure, bipolar soft exterior, bipolar soft boundary and establish some important properties on them. Some relations between them are also discussed. Moreover, the notions of bipolar soft point, bipolar soft limit point and the derived set of a bipolar soft set are discussed. In additions, examples are presented to illustrate our work.

Soft Sets and Its Applications

2016 ◽  
pp. 65-85 ◽  
Author(s):  
B. K. Tripathy ◽  
K. R. Arun
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Soft Set ◽  
Soft Sets ◽  
Practical Applications ◽  
Set Operations ◽  
New Models ◽  
Pros And Cons

Uncertainty is an inherent characteristic of modern day databases. In order to handle such databases with uncertainty, several new models have been introduced in the literature. Some new models like fuzzy sets introduced by Zadeh (1965), rough sets invented by Z. Pawlak (1982) and intuitionistic fuzzy sets extended by K.T. Atanassov (1986). All these models have their own pros and cons. However, one of the major problems with these models is the lack of sufficient number of parameters to deal with uncertainty. In order to add adequate number of parameters, soft set theory was introduced by Molodtsov in 1999. Since then the theoretical developments on soft set theory has attracted the attention of researchers. However, the practical applications of any theory are of enough importance to make use of it. In this chapter, the basic definitions of soft set, operations and properties are discussed. Also, the aim in this chapter is to discuss on the different applications of soft sets; like decision making, parameter reduction, data clustering and data dealing with incompleteness.

scholarly journals Generalized Trapezoidal Fuzzy Soft Set and Its Application in Medical Diagnosis

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Haidong Zhang ◽  
Lan Shu ◽  
Shilong Liao
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Medical Diagnosis ◽  
Soft Set ◽  
Mathematical Tool ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Decision Making Problem ◽  
Fuzzy Soft Sets

Soft set theory is a newly emerging mathematical tool to deal with uncertain problems. In this paper, by introducing a generalization parameter, which itself is trapezoidal fuzzy, we define generalized trapezoidal fuzzy soft sets and then study some of their properties. Finally, applications of generalized trapezoidal fuzzy soft sets in a decision making problem and medical diagnosis problem are shown.

scholarly journals Soft Rough Approximation Operators and Related Results

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Zhaowen Li ◽  
Bin Qin ◽  
Zhangyong Cai
Keyword(s):  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Soft Set ◽  
Soft Sets ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Generalized Rough Set ◽  
Special Case

Soft set theory is a newly emerging tool to deal with uncertain problems. Based on soft sets, soft rough approximation operators are introduced, and soft rough sets are defined by using soft rough approximation operators. Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model. This paper is devoted to investigating soft rough approximation operations and relationships among soft sets, soft rough sets, and topologies. We consider four pairs of soft rough approximation operators and give their properties. Four sorts of soft rough sets are investigated, and their related properties are given. We show that Pawlak's rough set model can be viewed as a special case of soft rough sets, obtain the structure of soft rough sets, give the structure of topologies induced by a soft set, and reveal that every topological space on the initial universe is a soft approximating space.

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