Another approach to soft rough sets
Introduction
Soft set theory is a new approach to deal with uncertainty. Prior to the inception of soft set theory, probability theory, fuzzy set and rough set theory were common tools to deal with uncertainty. Although these theories have been applied to many problems successfully yet there are certain difficulties associated with these theories [20]. For example in probability theory, in order to check the stability of the system a large number of experiments is required. In economics and environment sciences such an experimentation is not affordable. In case of fuzzy set theory defining a membership function is not an easy task. Theory of rough sets requires an equivalence relation among the members of the set under consideration. In many daily life situations such an equivalence relation is very difficult to find due to imprecise human knowledge. Perhaps above mentioned difficulties associated with these theories are due to lack of parametrization tools [20]. Theory of soft sets presented by Molodtsov has enough parameters, so that it is free from above mentioned difficulties. Soft set theory deals with uncertainty and vagueness on the one hand while on the other it has enough parametrization tools. These qualities of soft set theory make it popular among researchers and experts working in diverse areas. Applications of soft set theory can be seen in [3], [6], [11], [14], [17], [21], [30], [31], [33]. Theoretical aspects of soft sets are considered in [19]. Ali et al. introduced some new operations in soft set theory [1].
Theory of rough sets is presented by Pawlak [24]. In order to handle vagueness and imprecision in the data equivalence relations play an important role in this theory. It is well known that an equivalence relation on a set partitions the set into disjoint classes and vice versa. A subset which can be written as union of these classes is called definable otherwise it is not definable. In general a subset of a set is not definable, however it can be approximated by two definable subsets called lower and upper approximations of the set. These approximations provide us useful knowledge hidden in the data for decision making. This theory has been applied successfully to solve many problems, but in daily life, it is very difficult to find an equivalence relation among the elements of a set under consideration. Therefore some other rough set models are introduced. In [32] reflexive relation (such relations are called similarity relations) are considered to define a more general rough sets model. Some other general relations such as tolerance relations and dominance relations are considered to define rough sets models [12], [36], [38]. Concept of rough sets with cover is another generalization. In rough set theory with covers lower and upper approximations of a subset are determined with the help of elements of the cover [7], [9], [37]. Another generalization of rough sets is called T-rough sets, here a map T is utilized to find approximations of the set [5], [8], [35]. Applications of rough sets in various fields can be seen in [12], [15], [16], [23], [24], [25], [26], [29].
Rough set theory and soft set theory are two different tools to deal with uncertainty. Apparently there is no direct connection between these two theories, however efforts have been made to establish some kind of linkage [2], [6], [10]. The major criticism on rough set theory is that it lacks parametrization tools [19]. In order to make parametrization tools available in rough sets a major step is taken by Feng et al. in [10]. They introduce the concept of soft rough sets, where instead of equivalence classes parametrized subsets of a set serve the purpose of finding lower and upper approximations of a subset. In doing so, some unusual situations may occur. For example upper approximation of a nonempty set may be empty. Upper approximation of a subset X may not contain the set X. These situations does not occur in classical rough set theory. Therefore it is natural to ask, “Can we define a soft rough set model where these situations may not occur?” In the present paper we endeavour to find a positive answer of this question.
In order to strengthen the concept of soft rough sets a new approach is being presented here. Mathematically, this so called notion of modified soft rough sets (MSR-sets) and its lower and upper approximations may seem different from the classical rough set theory and soft set theory but the underlying concepts are very similar. MSR-sets satisfy all the basic properties of rough sets and parametrization aspect also remains valid. This paper is arranged in the following manner. In Section 2, some basic notions of rough sets and soft sets are given. Notion of soft rough sets presented by Feng et al. is also discussed here. Section 3, is devoted for the study of Modified soft rough sets. Certain types of relations originate from lower and upper approximations in MSR-sets. These are actually equivalence relations. Some properties of these relations are studied in Section 4. Accuracy measure for a subset gives us an idea how accurately a subset of the universe is approximated? In section 5, accuracy measures for MSR-sets and soft rough sets are defined and some of their properties are studied. In Section 6, concept of approximations of an information system with respect to another information system is studied.
Section snippets
Preliminaries
In this section, we recall definitions of rough set, soft set, soft rough set and some related concepts.
In rough set theory indiscernibility relation, generated by information about objects of interest are of basic importance. The indiscernibility relation expresses the fact that due to the lack of knowledge, we are unable to differentiate some objects by employing the available information. This means that, in general, we cannot deal with single objects but we have to consider clusters of
Modified soft rough sets (MSR-sets)
In the previous section, we have seen that by applying the definition of lower rough approximation and upper rough approximation for a subset of the universe set some basic properties of rough sets may evaporate and unfortunate elements which are in NegP(X) for all X ⊆ U, cannot be avoided. The underlying purpose of soft sets is the parametrization of subsets of universe U and of rough sets is to deal with vague concepts caused by indiscernibility in information. In the following we strengthen
Relations associated with lower and upper approximation in MSR-sets
In [10] relations associated with lower and upper approximations are studied. In this section it is shown that similar properties also hold for MSR-sets. Definition 16 Let (F, A) be a soft set over U and (U, φ) be an MSR-approximation space. Then for all X, Y ∈ P(U) defineThese binary relations may be called as lower MSR relation, upper MSR relation and lower–upper MSR relations respectively. Proposition 5 The relations and are
Accuracy measure for MSR-sets
In [23] Pawlak has narrated that some measures associated with rough sets can be helpful to get an idea, how accurate is the information related with some equivalence relation for a particular classification. Accuracy measure for different types of rough sets has been discussed in [8], [18], [34]. In this section, we study the measure of accuracy for soft rough sets and modified soft rough sets and study some of their properties.
By Pawlak [24] accuracy measure of an approximation space (U, σ
Modified rough approximations of soft sets
It is well known that an information system can be represented by a soft set and vice versa [4]. In this section we introduce the concept of approximations on an information system with respect to another information system. That is we study upper and lower MSR-approximations of soft sets with respect to another soft set.
Let (F, A) be a soft set defined over the universe set U. (U, φ) be the MSR-approximation space where φ :U → P(A) is defined as φ(x) = {a:x ∈ F(a)}. Let (G, B) be a soft set defined
Conclusions
Soft sets and rough sets are two remarkable theories. Attempts have been made to combine these two theories. During this effort certain shortcomings became the part of soft rough sets. In order to overcome these deficiencies Modified Soft Rough sets (MSR-sets) have been introduced in this paper. Some basic properties of MSR-sets have been investigated. Similar results which require some strong conditions for their proof in soft rough sets can be proved in MSR-sets without these conditions.
Acknowledgements
Authors are highly grateful to anonymous referees and Hamido Fujita, editor in chief, for their kind suggestions which helped to improve this paper.
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