A New Approach to Entropy and Similarity Measure of Vague Soft Sets

We focus our discussion on the uncertainty measures of vague soft sets. We propose axiomatic definitions of similarity measure and entropy for vague soft sets. Furthermore, we present a new category of similarity measures and entropies for vague soft sets. The basic properties of these measures are discussed and the relationships among these measures are analyzed.


Introduction
In 1999, Molodtsov [1] introduced the concept of soft sets, which can be considered as a new mathematical tool for dealing with uncertainties that traditional mathematical tools cannot handle. A soft set is a collection of approximate descriptions of an object. The absence of restriction on the approximate description in soft set theory makes it very convenient to apply. Recently, applications of soft sets have surged in various areas, including decision making, data analysis, simulation, and texture classification [2][3][4][5][6][7][8][9][10][11].
Accordingly, works on soft set theory are progressing rapidly. Maji et al. [12] defined several algebraic operations on soft sets and conducted a theoretical study on the theory of soft sets. Based on [12], Ali et al. [13] introduced some new operations on soft sets and improved the notion of complement of soft set. They proved that certain De Morgan's laws with respect to these new operations hold in soft set theory. Qin and Hong [14] introduced the notion of soft equality and established lattice structures and soft quotient algebras of soft sets. Maji et al. [15] initiated the study on hybrid structures involving soft sets and fuzzy sets. They proposed the notion of fuzzy soft set as a fuzzy generalization of classical soft sets and some basic properties were discussed. Afterwards, many researchers have worked on this concept. Various kinds of extended fuzzy soft sets such as generalized fuzzy soft sets [16], intuitionistic fuzzy soft sets [17,18], interval-valued fuzzy soft sets [19], vague soft sets [20], interval-valued intuitionistic fuzzy soft sets [21], and soft interval set [22] have been proposed. The combination of soft set and rough set [23] has also been extensively investigated [24][25][26][27].
The measurement of uncertainty is an important topic for the theories dealing with uncertainty. Majumdar and Samanta [28] initiated the study of uncertainty measures of soft sets, in which some similarity measures between soft sets were presented. Recently, some related works concerning the uncertainty measures of soft sets, fuzzy soft sets, intuitionistic fuzzy soft set, and vague soft set were presented [29][30][31][32]. Wang and Qu [32] introduced axiomatic definitions of entropy, similarity measure, and distance measure for vague soft sets and proposed some formulas to calculate them. This paper is devoted to a further discussion of uncertainty measures for vague soft set. We make an analysis of the uncertainty measures presented in [32] and point out some drawbacks in it. First, a vague soft set is a parameterized family of vague sets on the universe. Different vague soft sets may have different parameter sets. The entropy, similarity measure, and distance measure presented in [32] are actually partial measures in the sense that they take only the vague soft sets with the whole parameter set into account. Second, the axiomatic definition of entropy is not complete; that is, in some cases, the definition cannot guarantee a crisper vague soft set has a smaller entropy. We illustrate this with 2 The Scientific World Journal an example. Based on these observations, we propose a new axiomatic definition of entropy and present a new approach to construct the similarity measures and entropies for vague soft sets. The paper is organized as follows. In Section 2, we recall some notions and properties of soft sets and vague soft sets. In Section 3, we analyze the axiomatic definitions of similarity measure, distance measure, and entropy presented in [32] and point out some drawbacks in it. The new axiomatic definitions of similarity measure and entropy are presented. In Section 4, we propose a new approach to construct similarity measures between vague soft sets. Section 5 is devoted to the construction of entropy for vague soft set based on similarity measures. The paper is completed with some concluding remarks.

Overview of Soft Sets and Vague Soft Sets
In this section, we recall some fundamental notions of soft sets and vague soft sets. See especially [1,20,33,34] for further details and background.
The theory of fuzzy sets initiated by Zadeh [34] provides an appropriate framework for representing and processing vague concepts by allowing partial memberships. Let be a nonempty set, called universe. A fuzzy set on is defined by a membership function : For ∈ , the membership value ( ) essentially specifies the degree to which belongs to the fuzzy set . We denote by ( ) the set of all fuzzy sets on .
Among the extensions of the classic fuzzy set, vague set is one of the most popular sets treating imprecision and uncertainty. It was proposed by Gau and Buehrer [33]. indicates that the exact grade of membership of to may be unknown, but it is bounded by ( ) and 1 − ( ).
Every fuzzy set corresponds to the following vague set: Thus, the notion of vague sets is a generalization of fuzzy sets.
The operations on vague sets are natural generalizations of the corresponding operations on fuzzy sets. Also, the notion of vague subset is a generalization of the notion of fuzzy subset.
In 1999, Molodtsov [1] proposed a new concept called soft set to model uncertainties, which associates a set of objects with a set of parameters. Concretely, let be the universe set and the set of all possible parameters under consideration with respect to . Usually, parameters are attributes, characteristics, or properties of objects in . ( , ) will be called a soft space. Molodtsov defined the notion of a soft set in the following way.
Definition 4 (see [1]). A pair ( , ) is called a soft set over , where ⊆ and is a mapping given by : → ( ).
In other words, a soft set over is a parameterized family of subsets of . is called the parameter set of the soft set ( , ). For ∈ , ( ) may be considered as the set ofapproximate elements of ( , ). For illustration, we consider the following example of soft set.
In this case, to define a soft set means to point out expensive houses, beautiful houses, and so on. The soft set ( , ) may describe the "attractiveness of the houses" which Mr. X is going to buy. Suppose that Then the soft set ( , ) is a parameterized family { ( ); 1 ≤ ≤ 5} of subsets of and gives us a collection of approximate descriptions of an object. ( 1 ) = {ℎ 2 , ℎ 4 } means "houses ℎ 2 and ℎ 4 " are "expensive. " Maji et al. [15] initiated the study on hybrid structures involving soft sets and fuzzy sets. They proposed the notion of fuzzy soft set by combining soft sets and fuzzy sets. Afterwards, various kinds of extended fuzzy soft sets were presented. Xu et al. [20] proposed the notion of vague soft set as follows.
Definition 6 (see [20]). Let be an initial universe set, ( ) the set of all vague sets over , and a set of parameters. A pair ( , ) is called a vague soft set over , where ⊆ and is a mapping given by : → ( ).

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In what follows, we denote by VSS( ) the set of all vague soft sets over .
By the interpretation of true membership and false membership, a null vague soft set 0 is actually a soft set ( , ) satisfying ( ) = 0 for any ∈ and an absolute vague soft set is actually a soft set ( , ) satisfying ( ) = for any ∈ . ∈ . This definition is in essence equivalent to Definition 9. But by using this definition, it is not convenient when considering the grade of similarity between ( , ) and ( , ) , because the parameter sets of ( , ) and ( , ) are disjoint in mathematics; that is, ∩ ¬ = 0. In general, comparing approximate sets for different parameters is not reasonable.

Analysis of the Existing Entropy of Vague Soft Set
The measurement of uncertainty is an important topic for the theory dealing with uncertainty. The entropy, similarity measure, and distance measure in fuzzy set theory and the relationships among these measures have been extensively studied for their wide applications [35][36][37][38][39][40][41][42][43]. Wang and Qu [32] introduced the concepts of entropy, similarity measure, and distance measure of vague soft sets. Furthermore, the relationships among these measures were analyzed. In this section, we point out some drawbacks in [32] and present new definitions of entropy and similarity measure for vague soft sets.
Based on these observations, we introduce the new definition of entropy as follows. We note that the entropy presented in Theorem 3.1 [32] is also an entropy in the sense of Definition 13. Wang and Qu [32] proposed the axioms of similarity measure and distance measure for vague soft sets and discussed the relationships between these measures.
We notice that this theorem holds for the special similarity measure and distance measure presented in Theorems 3.2 and 3.3 [32]. But it does not hold in general; that is, one cannot prove (( , ), ( , )) + (( , ), ( , )) = 1 just by the definitions of similarity measure and distance measure. Furthermore, in Definitions 14 and 15, only the vague soft sets with the whole parameter set are compared. Thus the similarity measure and distance measure are all partial measures. By the way, conditions ( 2) and ( 2) are clearly not necessary because the codomain of and has already been restricted to [0, 1].
Similarity measure and distance measure are closely related. In what follows, we focus our discussion on entropy and similarity measure. Taking the above observations into account, we propose the following definition of similarity measure for vague soft sets.

Similarity Measures for Vague Soft Sets
Similarity measures quantify the extent to which different patterns, images, or sets are alike. In this section, we propose a new category of similarity measures (in the sense of Definition 17) for vague soft sets. Based on the notion of fuzzy equivalence proposed by Fodor and Roubens [44], Li et al. [37] proposed an approach to calculate the similarity degree between fuzzy sets. The approach can be summarized as the following theorem. Here some modifications on notations and technical terms have been made to fit the context of our discussion.
Theorem 18 (see [37]). Here, in order to avoid the denominator being zero, we set 0/0 = 1. By setting particular values of and , one can obtain some typical similarity measures for fuzzy sets [37]. Now we extended these measures to vague soft sets.
This completes the proof.
Proof. It can be proved in the same manner with Theorem 19.
is decreasing with respect to and . By it follows that ( , ) is increasing with respect to and . Thus we have the following corollary.

Concluding Remarks
Soft set theory was originally proposed as a general mathematical tool for dealing with uncertainties. Wang and Qu [32] introduced axiomatic definitions of entropy, similarity measure, and distance measure for vague soft sets and proposed some formulas to calculate them. This paper is devoted to a further discussion along this line. We point out that there are some drawbacks in [32] by examples. We propose a new axiomatic definition of entropy and present a new approach to construct the similarity measures and entropies for vague soft sets. Based on these uncertainty measures, we can further probe the applications of vague soft sets in the fields such as pattern recognition, data analysis, and decision making.