A New Approach to the Theory of Soft Sets

Mathematics

Soft set theory has evolved to provide a set of valuable tools for dealing with ambiguity and uncertainty in a variety of data structures related to real-world challenges. A soft set is characterized via a multivalued function of a set of parameters with certain conditions. In this study, we relax some conditions on the set of parameters and generalize some basic concepts in soft set theory. Specifically, we introduce generalized finite relaxed soft equality and generalized finite relaxed soft unions and intersections. The new operations offer a great improvement in the theory of soft sets in the sense of proper generalization and applicability.

European Journal of Pure and Applied Mathematics, 2016

In this paper, we study the theory of probabilistic soft sets introduced by Zhu and Wen \cite{ZhuWen}. We define equality of two probabilistic soft sets, subset, complement of a probabilistic soft set, impossible probabilistic soft set, certain probabilistic soft set with examples.We also introduce the operations of union, intersection, difference and symmetric difference. We prove that certain De Morgan's laws hold in probabilistic soft set theory with respect to these new definitions.

Global Trends in Intelligent Computing Research and Development

This chapter is about soft sets. A brief account of the developments that took place in last 14 years in the field of Soft Sets Theory (SST) has been presented. It begins with a brief introduction on soft sets and then it describes many generalizations of it. The notions of generalized fuzzy soft sets are defined and their properties are studied. After that, a notion of mapping, called soft mapping, in soft set setting is introduced. Later, algebraic structures on soft sets like soft group, soft ring, etc. are discussed. Then the next section deals with the concept of topology on soft sets. Here two notions of topology in soft sets are introduced, which are the topology of soft subsets and the soft topology, respectively. The idea of entropy for soft sets is defined in the later section. Next, some applications of hybrid soft sets in solving real life problems like medical diagnosis, decision-making, etc. are shown. Issues like measurement of similarity of soft sets are also addressed.

International Journal of Intelligent Systems and Applications, 2012

This paper aims to introduce the theory of imprecise soft sets which is a hybrid model of soft sets and imprecise sets. It has been established that two independent laws of randomness are necessary and sufficient to define a law of fuzziness. Further, in case of fuzzy sets, the set theoretic axioms of exclusion and contradiction are not satisfied. Accordingly, the theory of imprecise sets has been developed where these mistakes arising in the literature of fuzzy sets are absent. Our work is an endeavor to combine imprecise sets with soft sets resulting in imprecise soft sets. We have put forward a matrix representation of imprecise soft sets. Finally we have studied the notion of similarity of two imprecise soft sets and put forward an application of similarity in a decision problem.

Journal of New Theory, 2016

Many disciplines, including engineering, economics, medical science and social science are highly dependent on the task of modeling and computing uncertain data. When the uncertainty is highly complicated and difficult to characterize, classical mathematical approaches are often insufficient to derive effective or useful models. Testifying to the importance of uncertainties that cannot be defined by classical mathematics, researchers are introducing alternative theories every day. In addition to classical probability theory, some of the most important results on this topic are fuzzy sets, intuitionistic fuzzy sets, vague sets, interval-valued fuzzy set and rough sets. But each of these theories has its inherent limitations as pointed out by Molodtsov. For example, in probability theory, we require a large number of experiments in order to check the stability of the system. To define a membership function in case of fuzzy set theory is not always an easy task. Theory of rough sets requires an equivalence relation defined on the universal set under consideration. But in many real life situations such an equivalence relation is very difficult to find due to imprecise human knowledge. Perhaps the above mentioned difficulties associated with these theories are due to their incompatibility with the parameterization tools. Molodtsov introduced soft set theory as a completely new approach for modeling vagueness and uncertainty. This so-called soft set theory is free from the above mentioned difficulties as it has enough parameters. In soft set theory, the problem of setting membership function simply doesn't arise. This makes the theory convenient and easy to apply in practice. Soft set theory has potential applications in various fields including smoothness of functions, game theory, operations research, Riemann integration, probability theory and measurement theory. Most of these applications have already been demonstrated by Molodtsov. In this paper a new approach called refined soft sets is presented. Mathematically, this so called notion of refined soft sets may seem different from the classical soft set theory but the underlying concepts are very similar. In this paper the concept of refined soft set is introduced and the several operations between refined soft sets and soft sets are discussed. We also present the concept of soft images and soft inverse image of refined soft sets. The concept of image of a refined soft set has been used in a customer query problem.

Global Journal of Pure and Applied Mathematics, 2019

In this paper we present some of the main developments in the soft set theory as well as in the theory of algebraic structures and soft topology as a review of literature motivated by Molodsov.

2017

The aim of this paper is to exhibit the natural relation which exists between the Soft Set Theory and the f-Set Theory and in the end generalize the existing notions of 1. soft map 2. the soft image of a soft subset under a soft map and 3. the soft inverse image of a soft subset under a soft map and deduce several properties of these generalized soft images and generalized soft inverse images of the generalized soft subsets under generalized soft maps from the corresponding ones of f-maps.

International Journal on Cybernetics & Informatics, 2015

In this paper we define some new operations in fuzzy soft multi set theory and show that the De Morgan's type of results hold in fuzzy soft multi set theory with respect to these newly defined operations in our way. Also some new results along with illustrating examples have been put forward in our work.

In this paper, we study some operations of fuzzy soft sets and give fundamental properties of fuzzy soft sets. We discuss properties of fuzzy soft sets and their interrelation with respect to different operations such as union, intersection, restricted union and extended intersection. Then, we illustrate properties of OR, AND operations by giving counter examples. Also we prove that certain De Morgan's laws hold in fuzzy soft set theory with respect to different operations on fuzzy soft sets.

Journal of Mathematical and Computational Science, 2012

In this paper, we have defined disjunctive sum and difference of two fuzzy soft sets and study their basic properties. The notions of - cut soft set and - cut strong soft set of a fuzzy soft set have been put forward in our work. Some related properties have been established with proof, examples and counter examples.

Journal of Applied Mathematics, 2013

Using the notions of soft sets and N-structures, N-soft set theory is introduced. We apply it to both a decision making problem and a BCK/BCI algebra.

Hacettepe Journal of Mathematics and Statistics, 2017

In this paper, we introduce some new operations on type-2 soft sets and discuss related properties. The notions of primary empty type-2 soft sets, underlying empty type-2 soft sets and complete type-2 soft sets are introduced. In particular, we dene four new operations (the extension, the restriction, the extension-restriction, the restriction-extension) each on union, intersection and dierence. By using these new denitions we prove certain De Morgan's laws in type-2 soft set theory. Finally, an example which shows the validity of De Morgan's laws in real life problems is presented.

The aim of this paper is to exhibit the natural relation which exists between the Soft Set Theory and the f-Set Theory and in the end generalize the existing notions of 1. soft map 2. the soft image of a soft subset under a soft map and 3. the soft inverse image of a soft subset under a soft map and deduce several properties of these generalized soft images and generalized soft inverse images of the generalized soft subsets under generalized soft maps from the corresponding ones of f-maps.

With the integration of the concept of multisets in soft multiset theory in the literature, it is imperative that the algebraic properties of multisets be examined in the light of Soft multiset theory. To this end, a comprehensive study of these algebraic properties such as Commutative, Associative,Idempotent, Distributive, absorbtion and Demorgan’s laws are carried out. Though in general under this concept, the axiom of contradiction is satisfied with the exception of exclusion, we have established condition under which both are satisfied.

This paper studies some algebraic and lattice properties of soft sets. A soft binary operation is introduced and a few interesting results are investigated in this context.