Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 610125, 10 pages http://dx.doi.org/10.1155/2014/610125

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

Dan Hu,1 Zhiyong Hong,2 andYongWang1

1 Xihua University, Chengdu, Sichuan 610039, China 2 College of , Southwest Jiaotong University, Chengdu, Sichuan 610031, China

Correspondence should be addressed to Dan Hu; [email protected]

Received 22 March 2014; Revised 6 June 2014; Accepted 18 June 2014; Published 24 July 2014

Academic Editor: Feng Feng

Copyright © 2014 Dan Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We focus our discussion on the uncertainty measures of vague soft sets. We propose axiomatic 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.

1. Introduction -valued fuzzy soft sets19 [ ], vague soft sets20 [ ], interval-valued intuitionistic fuzzy soft sets [21], and soft In 1999, Molodtsov [1] introduced the concept of soft sets, interval [22]havebeenproposed.Thecombinationofsoft which can be considered as a new mathematical tool for set and rough set [23] has also been extensively investigated dealing with uncertainties that traditional mathematical tools [24–27]. cannot handle. A soft set is a collection of approximate The measurement of uncertainty is an important topic of an object. The absence of restriction on the for the theories dealing with uncertainty. Majumdar and approximate in soft makes it very Samanta [28]initiatedthestudyofuncertaintymeasuresof convenient to apply. Recently, applications of soft sets have soft sets, in which some similarity measures between soft sets surged in various areas, including decision making, data were presented. Recently, some related works concerning the analysis, simulation, and texture classification2 [ –11]. uncertainty measures of soft sets, fuzzy soft sets, intuitionistic Accordingly, works on soft set theory are progressing fuzzy soft set, and vague soft set were presented29 [ –32]. rapidly. Maji et al. [12] defined several algebraic operations Wang and Qu [32] introduced axiomatic definitions of on soft sets and conducted a theoretical study on the theory entropy, similarity measure, and distance measure for vague of soft sets. Based on12 [ ], Ali et al. [13]introducedsome soft sets and proposed some formulas to calculate them. new operations on soft sets and improved the notion of This paper is devoted to a further discussion of uncertainty of soft set. They proved that certain De Morgan’s measures for vague soft set. We make an analysis of the lawswithrespecttothesenewoperationsholdinsoftset uncertainty measures presented in [32] and point out some theory. Qin and Hong [14] introduced the notion of soft drawbacks in it. First, a vague soft set is a parameterized equality and established lattice structures and soft quotient family of vague sets on the universe. Different vague soft sets algebras of soft sets. Maji et al. [15]initiatedthestudyon may have different parameter sets. The entropy, similarity hybrid structures involving soft sets and fuzzy sets. They measure,anddistancemeasurepresentedin[32]areactually proposed the notion of fuzzy soft set as a fuzzy generalization partial measures in the sense that they take only the vague of classical soft sets and some basic properties were discussed. soft sets with the whole parameter set into account. Second, Afterwards, many researchers have worked on this concept. the axiomatic of entropy is not complete; that Various kinds of extended fuzzy soft sets such as generalized is, in some cases, the definition cannot guarantee a crisper fuzzy soft sets [16], intuitionistic fuzzy soft sets17 [ , 18], 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 𝐴 ∩ 𝐵 = {(𝑥,𝐴 [𝑡 (𝑥) ∧𝑡𝐵 (𝑥) ,(1−𝑓𝐴 (𝑥))∧(1−𝑓𝐵 (𝑥))]) ; axiomatic definition of entropy and present a new approach 𝑥∈𝑈} , to construct the similarity measures and entropies for vague soft sets. The paper is organized as follows. In Section 2,we 𝑐 𝐴 ={(𝑥,[𝑓𝐴 (𝑥) ,1−𝑡𝐴 (𝑥)]) ; 𝑥 ∈ 𝑈} . recall some notions and properties of soft sets and vague (2) soft sets. In Section 3, we analyze the axiomatic definitions of similarity measure, distance measure, and entropy presented 𝐴, 𝐵 in [32] and point out some drawbacks in it. The new axiomatic Definition 3 (see [33]). Let be two vague sets over the 𝑈 𝑥∈𝑈𝑡 (𝑥) ≤ 𝑡 (𝑥) 1−𝑓(𝑥) ≤ definitions of similarity measure and entropy are presented. universe .If,forall , 𝐴 𝐵 , 𝐴 1−𝑓(𝑥) 𝐴 𝐵 In Section 4, we propose a new approach to construct 𝐵 ,then is called a vague subset of , denoted by 𝐴⊆𝐵 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 The operations on vague sets are natural generalizations concluding remarks. of the corresponding operations on fuzzy sets. Also, the notion of vague subset is a generalization of the notion of fuzzy subset. 2. Overview of Soft Sets and Vague Soft Sets In 1999, Molodtsov [1]proposedanewconceptcalled soft set to model uncertainties, which associates a set of In this section, we recall some fundamental notions of soft objects with a set of parameters. Concretely, let 𝑈 be the sets and vague soft sets. See especially1 [ , 20, 33, 34]forfurther universe set and 𝐸 the set of all possible parameters under details and background. consideration with respect to 𝑈. Usually, parameters are The theory of fuzzy sets initiated by Zadeh34 [ ] provides attributes, characteristics, or properties of objects in 𝑈. (𝑈, 𝐸) an appropriate framework for representing and processing will be called a soft space. Molodtsov defined the notion of a vague concepts by allowing partial memberships. Let 𝑈 be a soft set in the following way. nonempty set, called universe. A 𝜇 on 𝑈 is defined by a membership 𝜇:𝑈 → [0,1].For𝑥∈𝑈, (𝐹, 𝐴) 𝑈 𝜇(𝑥) Definition 4 (see [1]). A pair is called a soft set over , the membership value essentially specifies the degree to where 𝐴⊆𝐸and 𝐹 is a mapping given by 𝐹 : 𝐴 → .𝑃(𝑈) which 𝑥 belongs to the fuzzy set 𝜇. We denote by 𝐹(𝑈) the set 𝑈 of all fuzzy sets on . In other words, a soft set over 𝑈 is a parameterized family Among the extensions of the classic fuzzy set, vague set of subsets of 𝑈. 𝐴 is called the parameter set of the soft set is one of the most popular sets treating imprecision and (𝐹, 𝐴).For𝑒∈𝐴, 𝐹(𝑒) may be considered as the set of 𝑒- uncertainty. It was proposed by Gau and Buehrer [33]. approximate elements of (𝐹, 𝐴). For illustration, we consider the following example of soft set. Definition 1 (see [33]). A vague set 𝐴 over the universe 𝑈 can 𝐴={(𝑥,[𝑡 (𝑥), 1 − 𝑓 (𝑥)]); 𝑥 ∈ be expressed by the notion 𝐴 𝐴 Example 5. Supposethattherearesixhousesintheuniverse 𝑈} 𝐴(𝑥) = [𝑡 (𝑥), 1 − 𝑓 (𝑥)] ;thatis, 𝐴 𝐴 , and the condition 𝑈 given by 𝑈={ℎ1,ℎ2,ℎ3,ℎ4,ℎ5,ℎ6} and 𝐸={𝑒1,𝑒2,𝑒3,𝑒4,𝑒5} 0≤𝑡(𝑥) ≤ 1 − 𝑓 (𝑥) 𝑥∈𝑈 𝐴 𝐴 should hold for any ,where is the set of parameters. 𝑒1, 𝑒2, 𝑒3, 𝑒4,and𝑒5 stand for the 𝑡 (𝑥) 𝐴 is called the membership degree (true membership) of parameters “expensive,” “beautiful,” “wooden,” “cheap,” and 𝑥 𝐴 𝑓 (𝑥) to the vague set , while 𝐴 is the degree of “inthegreensurroundings,”respectively. 𝑥 nonmembership (false membership) of the element to the In this case, to define a soft set means to point out 𝐴 vague set . expensive houses, beautiful houses, and so on. The soft set (𝐹, 𝐸) may describe the “attractiveness of the houses” which In this definition, 𝑡𝐴(𝑥) isalowerboundonthegradeof Mr. X is going to buy. Suppose that 𝐹(𝑒1)={ℎ2,ℎ4}, 𝐹(𝑒2)= membership of 𝑥 to 𝐴 derived from the evidence for 𝑥 and {ℎ1,ℎ3}, 𝐹(𝑒3)={ℎ3,ℎ4,ℎ5}, 𝐹(𝑒4)={ℎ1,ℎ3,ℎ5},and 𝑓𝐴(𝑥) is a lower bound on the negation of 𝑥 derived from 𝐹(𝑒5)={ℎ1}.Thenthesoftset(𝐹, 𝐸) is a parameterized family the evidence against 𝑥.Thevaguevalue[𝑡𝐴(𝑥), 1 −𝐴 𝑓 (𝑥)] {𝐹(𝑒𝑖); 1 ≤ 𝑖 ≤ 5} of subsets of 𝑈 and gives us a collection of indicates that the exact grade of membership of 𝑥 to 𝐴 may approximate descriptions of an object. 𝐹(𝑒1)={ℎ2,ℎ4} means be unknown, but it is bounded by 𝑡𝐴(𝑥) and 1−𝑓𝐴(𝑥). “houses ℎ2 and ℎ4” are “expensive.” Every fuzzy set 𝜇 corresponds to the following vague set: 𝜇 = {(𝑥, [𝜇 (𝑥) ,1−𝜇(𝑥)]) ; 𝑥 ∈ 𝑈} . Maji et al. [15]initiatedthestudyonhybridstructures (1) involving soft sets and fuzzy sets. They proposed the notion of fuzzy soft set by combining soft sets and fuzzy sets. Thus, the notion of vague sets is a generalization of fuzzy sets. Afterwards, various kinds of extended fuzzy soft sets were Definition 2 (see [33]). Let 𝐴, 𝐵 be two vague sets over the presented. Xu et al. [20] proposed the notion of vague soft universe 𝑈. The , intersection, and complement of set as follows. vague sets are defined as follows: Definition 6 (see [20]). Let 𝑈 be an initial universe set, 𝑉(𝑈) the set of all vague sets over 𝑈,and𝐸 a set of parameters. A 𝐴 ∪ 𝐵 = {(𝑥,𝐴 [𝑡 (𝑥) ∨𝑡𝐵 (𝑥) ,(1−𝑓𝐴 (𝑥))∨(1−𝑓𝐵 (𝑥))]) ; pair (𝐹, 𝐴) is called a vague soft set over 𝑈,where𝐴⊆𝐸and 𝑥∈𝑈} , 𝐹 is a mapping given by 𝐹 : 𝐴 → .𝑉(𝑈) The Scientific World Journal 3

In what follows, we denote by VSS(𝑈) the set of all vague (H2) 𝐻(𝐹, 𝐸) =1⇔ for all 𝑒∈𝐸, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) = soft sets over 𝑈. 𝑓𝐹(𝑒)(𝑥); 𝑐 Definition 7 (see [20]). For two vague soft sets (𝐹, 𝐴) and (H3) 𝐻(𝐹, 𝐸) = 𝐻((𝐹, 𝐸) ); (𝐺, 𝐵) over a universe 𝑈,onesaysthat(𝐹, 𝐴) is a vague soft (H4) for all 𝑒∈𝐸, 𝑥∈𝑈,when(𝐹, 𝐸) ⊆ (𝐺, 𝐸) and subset of (𝐺, 𝐵),if𝐴⊆𝐵and for all 𝑒∈𝐴, 𝐹(𝑒) is a vague 𝑡𝐺(𝑒)(𝑥) ≤𝐺(𝑒) 𝑓 (𝑥),or(𝐹, 𝐸) ⊇ (𝐺, 𝐸) and 𝑡𝐺(𝑒)(𝑥) ≥ subset of 𝐺(𝑒). This relation is denoted by (𝐹, 𝐴) ⊆ (𝐺,. 𝐵) 𝑓𝐺(𝑒)(𝑥),then𝐻(𝐹, 𝐸) ≤ 𝐻(𝐺, 𝐸). Definition 8 (see [20]). A vague soft set (𝐹, 𝐴) over 𝑈 is said In this definition, (𝐻1) meanstheentropyofasoftsetis to be a null vague soft set, denoted by 0𝐴,if𝑡𝐹(𝑒)(𝑥) =, 0 1− minimal. By (𝐻2), the entropy of the most vague soft set is 𝑓𝐹(𝑒)(𝑥) = 0 for any 𝑒∈𝐴and 𝑥∈𝑈. maximal. (𝐻3) means the entropies of a vague soft set and Avaguesoftset(𝐹, 𝐴) over 𝑈 is said to be an absolute its complement are equal. (𝐻1), (𝐻2),and(𝐻3) are natural vague soft set, denoted by 𝑈𝐴,if𝑡𝐹(𝑒)(𝑥) =, 1 1−𝑓𝐹(𝑒)(𝑥) = 1 generalizations of the corresponding conditions needed for for any 𝑒∈𝐴and 𝑥∈𝑈. the entropy of vague set. Now we focus our discussion on the (𝐻4) (𝐹, 𝐸) ⊆ (𝐺, 𝐸) 𝑡 (𝑥) ≤ 𝑓 (𝑥) By the interpretation of true membership and false condition .If , 𝐺(𝑒) 𝐺(𝑒) for any 0 (𝐹, 𝐴) 𝑒∈𝐸, 𝑥∈𝑈,then𝑡𝐺(𝑒)(𝑥) ≤ 0.5 by 𝑡𝐺(𝑒)(𝑥) ≤ 1 −𝐺(𝑒) 𝑓 (𝑥). membership,anullvaguesoftset 𝐴 is actually a soft set (𝐹, 𝐸) ⊆ (𝐺, 𝐸) 𝐹(𝑒) 𝐺(𝑒) satisfying 𝐹(𝑒) =0 for any 𝑒∈𝐴and an absolute vague soft In this case, means is crisper than 𝑈 (𝐹, 𝐴) 𝐹(𝑒) =𝑈 for any 𝑒∈𝐸.Similarly,if(𝐹, 𝐸) ⊇ (𝐺,, 𝐸) 𝑡𝐺(𝑒)(𝑥) ≥𝐺(𝑒) 𝑓 (𝑥) set 𝐴 is actually a soft set satisfying for any 𝑒∈𝐸𝑥∈𝑈 𝐹(𝑒) 𝐺(𝑒) 𝑒∈𝐴. for any , ,then is crisper than for any 𝑒∈𝐸.Thus(𝐻4) is reasonable, but it is not complete. In fact, (𝐹, 𝐸) (𝐺, 𝐸) Definition 9. The complement of a vague soft set (𝐹, 𝐴) is let be crisper that . There may exist parameters 𝑐 𝑐 𝑐 𝑒 ,𝑒 ∈𝐸 𝐹(𝑒 )⊆𝐺(𝑒) 𝐹(𝑒 )⊇𝐺(𝑒) denoted by (𝐹, 𝐴) =(𝐹,𝐴),where𝐹 : 𝐴 → 𝑉(𝑈) 1 2 such that 1 1 and 2 2 .Also,for 𝑒∈𝐸 𝑥∈ is a mapping given by 𝑡𝐹𝑐(𝑒)(𝑥) =𝐹(𝑒) 𝑓 (𝑥), 1−𝑓𝐹𝑐(𝑒)(𝑥) = a specific parameter , there may exist some elements 𝑈 ⊂𝑈 𝑡 (𝑥) ≤ 𝑓 (𝑥) 𝑡 (𝑥) ≤ 𝑡 (𝑥) 1−𝑡𝐹(𝑒)(𝑥) for any 𝑒∈𝐴and 𝑥∈𝑈. 1 such that 𝐺(𝑒) 𝐺(𝑒) , 𝐹(𝑒) 𝐺(𝑒) ,and 𝑓𝐹(𝑒)(𝑥) ≥𝐺(𝑒) 𝑓 (𝑥),and𝑡𝐺(𝑒)(𝑥) ≥𝐺(𝑒) 𝑓 (𝑥), 𝑡𝐹(𝑒)(𝑥) ≥𝐺(𝑒) 𝑡 (𝑥), Remark 10. In [32], the complement of a vague soft set (𝐹, 𝐴) and 𝑓𝐹(𝑒)(𝑥) ≤𝐺(𝑒) 𝑓 (𝑥) for any 𝑥 ∈ 𝑈−𝑈1. In these cases, (𝐻4) 𝑐 𝑐 is defined by (𝐹, 𝐴) =(𝐹,¬𝐴),where¬𝐴 = {¬𝑒; 𝑒, ∈𝐴} cannot guarantee 𝐻(𝐹, 𝐸) ≤ 𝐻(𝐺, 𝐸) because (𝐹, 𝐸) ⊆(𝐺,𝐸)̸ 𝑐 (𝐺, 𝐸) ⊆(𝐹,𝐸)̸ 𝐹 : ¬𝐴 → 𝑉(𝑈),and𝑡𝐹𝑐(¬𝑒)(𝑥) =𝐹(𝑒) 𝑓 (𝑥), 1−𝑓𝐹𝑐(¬𝑒)(𝑥) = and . As illustration, we consider the following 1−𝑡𝐹(𝑒)(𝑥) for any 𝑒∈𝐴and 𝑥∈𝑈. This definition example. is in essence equivalent to Definition 9. But by using this 𝐸={𝑒,𝑒 } 𝑈={𝑥} definition, it is not convenient when considering the grade of Example 12. (1) Let 1 2 , 1 .Thevaguesoftsets 𝑐 (𝐹, 𝐸) (𝐺, 𝐸) similarity between (𝐹, 𝐴) and (𝐹, 𝐴) , because the parameter and are defined by 𝑐 sets of (𝐹, 𝐴) and (𝐹, 𝐴) are disjoint in mathematics; that is, 𝐴∩¬𝐴=. 0 In general, comparing approximate sets for 𝐹(𝑒1)=(𝑥1, [0.2, 0.3]), 𝐹(𝑒2)=(𝑥1, [0.7, 0.8]), different parameters is not reasonable. (3) 𝐺(𝑒1)=(𝑥1, [0.4, 0.5]), 𝐺(𝑒2)=(𝑥1, [0.6, 0.7]).

3. Analysis of the Existing Entropy of By the interpretation of true membership and false mem- Vague Soft Set bership, 𝐹(𝑒1) and 𝐹(𝑒2) are crisper than 𝐺(𝑒1) and 𝐺(𝑒2), respectively. We noticed that (𝐹, 𝐸) ⊆(𝐺,𝐸)̸ and (𝐺, 𝐸) ⊈ The measurement of uncertainty is an important topic for (𝐹, 𝐸).Thus(𝐻4) can not guarantee 𝐻(𝐹, 𝐸) ≤ 𝐻(𝐺, 𝐸). the theory dealing with uncertainty. The entropy, similarity (2) Let 𝐸={𝑒}, 𝑈={𝑥1,𝑥2}.Thevaguesoftsets(𝐹, 𝐸) measure, and distance measure in fuzzy set theory and the and (𝐺, 𝐸) are defined by relationships among these measures have been extensively studied for their wide applications [35–43]. Wang and Qu 𝐹 (𝑒) =(𝑥, [0.2, 0.3])+(𝑥 , [0.7, 0.8]), [32] introduced the concepts of entropy, similarity measure, 1 2 (4) anddistancemeasureofvaguesoftsets.Furthermore,the 𝐺 (𝑒) =(𝑥, [0.4, 0.5])+(𝑥 , [0.6, 0.7]). relationships among these measures were analyzed. In this 1 2 section, we point out some drawbacks in [32] and present new definitions of entropy and similarity measure for vague soft Similarly, 𝐹(𝑒) is crisper than 𝐺(𝑒).Also,(𝐹, 𝐸) ⊆(𝐺,𝐸)̸ sets. and (𝐺, 𝐸) ⊆(𝐹,𝐸)̸ . (𝐻4) can not guarantee 𝐻(𝐹, 𝐸) ≤ 𝐻(𝐺,. 𝐸) Definition 11 (see [32]). Let 𝐻:VSS(𝑈) → [0, 1] be a mapp- ing. For (𝐹, 𝐸) ∈ VSS(𝑈), 𝐻(𝐹, 𝐸) is called the entropy of By the way, VSS(𝑈) is the set of all vague soft sets over 𝑈, (𝐹, 𝐸) if it satisfies the following conditions: but (H1)∼(H4) are all talking about the vague soft sets with the whole parameter set 𝐸. Thus, the entropy derived from this definition is actually a partial entropy. (H1) 𝐻(𝐹, 𝐸) =0⇔ for all 𝑒∈𝐸, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =, 0 Basedontheseobservations,weintroducethenewdefini- 𝑓𝐹(𝑒)(𝑥) =,or 1 𝑡𝐹(𝑒)(𝑥) =, 1 𝑓𝐹(𝑒)(𝑥) =; 0 tion of entropy as follows. 4 The Scientific World Journal

Definition 13. Let 𝐻:VSS(𝑈) → [0, 1] be a mapping. For We notice that this theorem holds for the special simi- (𝐹, 𝐴) ∈ VSS(𝑈), 𝐻(𝐹, 𝐴) is called the entropy of (𝐹, 𝐴) if it larity measure and distance measure presented in Theorems satisfies the following conditions: 3.2 and 3.3 [32].Butitdoesnotholdingeneral;thatis,one cannot prove 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) + 𝑑((𝐹, 𝐸), (𝐺,𝐸))=1 just by (H1) 𝐻(𝐹, 𝐴) =0⇔ for all 𝑒∈𝐴, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =, 0 thedefinitionsofsimilaritymeasureanddistancemeasure. 𝑓𝐹(𝑒)(𝑥) =,or 1 𝑡𝐹(𝑒)(𝑥) =, 1 𝑓𝐹(𝑒)(𝑥) =; 0 Furthermore, in Definitions 14 and 15,onlythevaguesoft 𝐻(𝐹, 𝐴) =1⇔ 𝑒∈𝐴𝑥∈𝑈𝑡 (𝑥) = (H2) for all , , 𝐹(𝑒) sets with the whole parameter set 𝐸 are compared. Thus 𝑓 (𝑥) 𝐹(𝑒) ; the similarity measure and distance measure are all partial 𝑐 (H3) 𝐻(𝐹, 𝐴) = 𝐻((𝐹, 𝐴) ); measures. By the way, conditions (𝑀2) and (𝑑2) are clearly 𝑀 𝑑 (H4) Let (𝐹, 𝐴), (𝐺, 𝐴)∈ VSS(𝑈).Ifforall𝑒∈𝐴, 𝑥∈ not necessary because the codomain of and has already been restricted to [0, 1]. 𝑈, 𝑡𝐹(𝑒)(𝑥) ≤𝐺(𝑒) 𝑡 (𝑥), 𝑓𝐹(𝑒)(𝑥) ≥𝐺(𝑒) 𝑓 (𝑥) whenever Similarity measure and distance measure are closely 𝑡𝐺(𝑒)(𝑥) ≤𝐺(𝑒) 𝑓 (𝑥),and𝑡𝐹(𝑒)(𝑥) ≥𝐺(𝑒) 𝑡 (𝑥), 𝑓𝐹(𝑒)(𝑥) ≤ related. In what follows, we focus our discussion on entropy 𝑓𝐺(𝑒)(𝑥) whenever 𝑡𝐺(𝑒)(𝑥) ≥𝐺(𝑒) 𝑓 (𝑥),then𝐻(𝐹, 𝐴) ≤ 𝐻(𝐺,. 𝐴) and similarity measure. Taking the above observations into account, we propose the following definition of similarity We note that the entropy presented in Theorem 3.132 [ ]is measureforvaguesoftsets. also an entropy in the sense of Definition 13.WangandQu 𝑀: (𝑈) × (𝑈) → [0, 1] [32] proposed the of similarity measure and distance Definition 17. Let VSS VSS be a map- ping. For (𝐹, 𝐴) ∈ VSS(𝑈) and (𝐺, 𝐵) ∈ VSS(𝑈), 𝑀((𝐹, 𝐴), measureforvaguesoftsetsanddiscussedtherelationships (𝐺, 𝐵)) (𝐹, 𝐴) between these measures. is called the degree of similarity between and (𝐺, 𝐵) if it satisfies the following conditions: Definition 14 (Definition 3.2,32 [ ]). Let 𝑀:VSS(𝑈) × (M1) 𝑀((𝐹, 𝐴), (𝐺, 𝐵)) = 𝑀((𝐺, ;𝐵), (𝐹,𝐴)) VSS(𝑈) → [0, 1] be a mapping. For (𝐹, 𝐸) ∈ VSS(𝑈) and (M2) 𝑀((𝐹, 𝐴), (𝐺, 𝐵)) = 1 ⇔ (𝐹, 𝐴)=(𝐺,𝐵); (𝐺, 𝐸) ∈ VSS(𝑈), 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) is called the degree of 𝑀((𝐹, 𝐴), (𝐺, 𝐵)) =0⇔ 𝑒∈𝐴∩𝐵(𝐴∩𝐵≠ similarity between (𝐹, 𝐸) and (𝐺, 𝐸) if it satisfies the following (M3) for all 0) 𝑥∈𝑈𝑡 (𝑥) = 0 𝑓 (𝑥) = 1 𝑡 (𝑥) = 1 conditions: , , 𝐹(𝑒) , 𝐹(𝑒) , 𝐺(𝑒) , 𝑓𝐺(𝑒)(𝑥) =,or 0 𝑡𝐹(𝑒)(𝑥) =, 1 𝑓𝐹(𝑒)(𝑥) =, 0 𝑡𝐺(𝑒)(𝑥) = 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) = 𝑀((𝐺, 𝐸), (𝐹,𝐸)) (M1) ; 0, 𝑓𝐺(𝑒)(𝑥) =; 1 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) ∈[0,1] (M2) ; (M4) (𝐹, 𝐴) ⊆ (𝐺, 𝐵) ⊆(𝑃,𝐶) implies 𝑀((𝐹, 𝐴), (𝑃, 𝐶))≤ (M3) 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) = 1 ⇔ (𝐹,; 𝐸)=(𝐺, min(𝑀((𝐹, 𝐴), (𝐺, 𝐵)), 𝑀((𝐺,. 𝐵),(𝑃,𝐶))) (M4) 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) =0⇔ for all 𝑒∈𝐸, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =, 0 𝑓𝐹(𝑒)(𝑥) =, 1 𝑡𝐺(𝑒)(𝑥) =, 1 𝑓𝐺(𝑒)(𝑥) =,or 0 4. Similarity Measures for Vague Soft Sets 𝑡𝐹(𝑒)(𝑥) =, 1 𝑓𝐹(𝑒)(𝑥) =, 0 𝑡𝐺(𝑒)(𝑥) =, 0 𝑓𝐺(𝑒)(𝑥) =; 1 Similarity measures quantify the extent to which different (M5) (𝐹, 𝐸) ⊆ (𝐺, 𝐸) ⊆(𝑃,𝐸) implies 𝑀((𝐹, 𝐸), (𝑃, 𝐸))≤ patterns,images,orsetsarealike.Inthissection,wepropose min(𝑀((𝐹, 𝐸), (𝐺, 𝐸)), 𝑀((𝐺,. 𝐸),(𝑃,𝐸))) a new category of similarity measures (in the sense of Definition 15 (Definition 3.2,32 [ ]). Let 𝑑:VSS(𝑈) × Definition 17) for vague soft sets. VSS(𝑈) → [0, 1] be a mapping. For (𝐹, 𝐸) ∈ VSS(𝑈) and Basedonthenotionoffuzzyequivalenceproposedby (𝐺, 𝐸) ∈ VSS(𝑈), 𝑑((𝐹, 𝐸), (𝐺, 𝐸)) is called the degree of Fodor and Roubens [44], Li et al. [37] proposed an approach distance between (𝐹, 𝐸) and (𝐺, 𝐸) if it satisfies the following to calculate the similarity degree between fuzzy sets. The conditions: approach can be summarized as the following theorem. Here some modifications on notations and technical terms have (d1) 𝑑((𝐹, 𝐸), (𝐺, 𝐸)) = 𝑑((𝐺, ;𝐸), (𝐹,𝐸)) been made to fit the context of our discussion. 𝑑((𝐹, 𝐸), (𝐺, 𝐸)) ∈[0,1] (d2) ; Theorem 18 𝑁 𝑁 𝑑((𝐹, 𝐸), (𝐺, 𝐸)) =1⇔ 𝑒∈𝐸𝑥∈𝑈 (see [37]). Suppose 𝜃 and 𝛿 are functions (d3) for all , , defined for all 𝐴, 𝐵 ∈ 𝐹(𝑈) by 𝑡𝐹(𝑒)(𝑥) =, 0 𝑓𝐹(𝑒)(𝑥) =, 1 𝑡𝐺(𝑒)(𝑥) =, 1 𝑓𝐺(𝑒)(𝑥) =,or 0 𝑡𝐹(𝑒)(𝑥) =, 1 𝑓𝐹(𝑒)(𝑥) =, 0 𝑡𝐺(𝑒)(𝑥) =, 0 𝑓𝐺(𝑒)(𝑥) =; 1 𝑁𝜃 (𝐴,) 𝐵 𝑑((𝐹, 𝐸), (𝐺, 𝐸)) = 0 ⇔ (𝐹, 𝐸)=(𝐺, (d4) ; = ∑ (𝑎−𝑎|𝐴 (𝑥) −𝐵(𝑥)| (d5) (𝐹, 𝐸) ⊆ (𝐺, 𝐸) ⊆(𝑃,𝐸) implies 𝑑((𝐹, 𝐸), (𝑃, 𝐸))≥ 𝑥∈𝑈 max(𝑑((𝐹, 𝐸), (𝐺, 𝐸)), 𝑑((𝐺,. 𝐸),(𝑃,𝐸))) +𝑏⋅min (𝐴 (𝑥) ,𝐵(𝑥))) Theorem 16 (see [32]). Let 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) and 𝑑((𝐹, 𝐸), (𝐺, 𝐸)) be the similarity measure and distance measure ×(∑ (𝑎−(𝑎−1) |𝐴 (𝑥) −𝐵(𝑥)| (𝐹, 𝐸) (𝐺, 𝐸) between two vague soft sets and as defined in 𝑥∈𝑈 Definitions 13 and 14, respectively. Then the relations between 𝑀((𝐹, 𝐸), (𝐺, 𝐸)) and 𝑑((𝐹, 𝐸), (𝐺, 𝐸)) can be given as follows: −1 +𝑏⋅min (𝐴 (𝑥) ,𝐵(𝑥))) ) , 𝑀 ((𝐹, 𝐸) , (𝐺, 𝐸)) +𝑑((𝐹, 𝐸) , (𝐺, 𝐸)) =1. (5) The Scientific World Journal 5

𝑁 (𝐴,) 𝐵 󵄨 󵄨 𝛿 > ∑ (𝑎 − 𝑎 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨 𝑥∈𝑈 = ∑ (𝑎−𝑎|𝐴 (𝑥) −𝐵(𝑥)| 𝑥∈𝑈 +𝑏⋅min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐹(𝑒) (𝑥))) +𝑏(1−max (𝐴 (𝑥) ,𝐵(𝑥)))) (8)

×(∑ (𝑎−(𝑎−1) |𝐴 (𝑥) −𝐵(𝑥)| and consequently, 𝑁𝜃(𝑡𝐹(𝑒),𝑡𝐺(𝑒))<1. This is a contradiction. 𝑥∈𝑈 −1 Thus 𝑡𝐹(𝑒)(𝑥) =𝐺(𝑒) 𝑡 (𝑥) for each 𝑒∈𝐴, 𝑥∈𝑈.Similarly,we 𝑓 (𝑥) = 𝑓 (𝑥) 𝑒∈𝐴,𝑥∈𝑈 + 𝑏(1 − max(𝐴(𝑥), 𝐵(𝑥)))) ) have 𝐹(𝑒) 𝐺(𝑒) for each .Consequently, we have (𝐹, 𝐴) = (𝐺,. 𝐵) (6) (M3) Let 𝑀((𝐹, 𝐴), (𝐺,.Itfollowsthat,foreach 𝐵))=0 𝑒∈𝐴∩𝐵, 𝑁𝜃(𝐹(𝑒), 𝐺(𝑒)) =0 and hence 𝑁𝜃(𝑡𝐹(𝑒),𝑡𝐺(𝑒))=0, with 𝑎≥0, 𝑏≥0,and𝑎+𝑏;then, >0 𝑁𝜃 and 𝑁𝛿 are similarity 𝑁𝜃(𝑓𝐹(𝑒),𝑓𝐺(𝑒))=0.By𝑁𝜃(𝑡𝐹(𝑒),𝑡𝐺(𝑒))=0we have, for each measures for fuzzy sets in the sense that 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =, 0 𝑡𝐺(𝑒)(𝑥) = 1 or 𝑡𝐹(𝑒)(𝑥) =, 1 𝑡𝐺(𝑒)(𝑥) =. 0 (1) 𝑁𝜃(𝑈, 0) =, 0 𝑁𝛿(𝑈, 0) = 0 and 𝑁𝜃(𝐴, 𝐴), =1 𝑁𝛿(𝐴, 𝐴) =1 whenever 𝐴∈𝐹(𝑈); (1) If 𝑡𝐹(𝑒)(𝑥) =, 0 𝑡𝐺(𝑒)(𝑥) =,then 1 𝑓𝐺(𝑒)(𝑥) = 0 by (2) 𝑁𝜃(𝐴, 𝐵)𝜃 =𝑁 (𝐵, 𝐴), 𝑁𝛿(𝐴, 𝐵)𝛿 =𝑁 (𝐵, 𝐴) whenever 𝑡𝐺(𝑒)(𝑥) +𝐺(𝑒) 𝑓 (𝑥) ≤.Thus 1 𝑡𝐺(𝑒)(𝑥) = 1 by 𝑁𝜃(𝑓𝐹(𝑒), 𝐴, 𝐵 ∈ 𝐹(𝑈); 𝑓𝐺(𝑒))=0. (3) for all 𝐴, 𝐵, 𝐶 ∈ 𝐹(𝑈), 𝑁𝜃(𝐴, 𝐶) ≤ min(𝑁𝜃(𝐴, 𝐵), 𝑁𝜃(𝐵, 𝐶)), 𝑁𝛿(𝐴, 𝐶) ≤ min(𝑁𝛿(𝐴, 𝐵),𝛿 𝑁 (𝐵, 𝐶)) (2) If 𝑡𝐹(𝑒)(𝑥) =, 1 𝑡𝐺(𝑒)(𝑥) =,then 0 𝑓𝐹(𝑒)(𝑥) = 0 by 𝑡𝐹(𝑒) whenever 𝐴⊆𝐵⊆𝐶. (𝑥) +𝐹(𝑒) 𝑓 (𝑥) ≤. 1 Therefore 𝑓𝐺(𝑒)(𝑥) = 1 by 𝑁 (𝑓 ,𝑓 )=0 Here, in order to avoid the denominator being zero, we set 𝜃 𝐹(𝑒) 𝐺(𝑒) . 0/0 = 1. By setting particular values of 𝑎 and 𝑏,onecanobtain some typical similarity measures for fuzzy sets [37]. Now we The converse implication is trivial. extended these measures to vague soft sets. (M4) Let (𝐹, 𝐴), (𝐺, 𝐵), (𝑃,𝐶)∈ VSS(𝑈) and (𝐹, 𝐴) ⊆ (𝐺, 𝐵) ⊆ (𝑃,.Itfollowsthat 𝐶) 𝐴⊆𝐵⊆𝐶and 𝐹(𝑒) ⊆ 𝐺(𝑒) Theorem 19. 𝑀𝜃 : 𝑉𝑆𝑆(𝑈) × 𝑉𝑆𝑆(𝑈) →[0,1] is a similarity ⊆ 𝑃(𝑒) 𝑒∈𝐴 𝑡 ⊆𝑡 ⊆𝑡 measure, where, for any (𝐹, 𝐴), (𝐺, 𝐵) ∈, 𝑉𝑆𝑆(𝑈) for each .Consequently, 𝐹(𝑒) 𝐺(𝑒) 𝑃(𝑒), 𝑓 ⊆𝑓 ⊆𝑓 𝑁 (𝑡 ,𝑡 )≤𝑁(𝑡 , 1 𝑃(𝑒) 𝐺(𝑒) 𝐹(𝑒).Thuswehave 𝜃 𝐹(𝑒) 𝑃(𝑒) 𝜃 𝐹(𝑒) 𝑀 ((𝐹, 𝐴) , (𝐺,)) 𝐵 = ∑ 𝑁 (𝐹 (𝑒) ,𝐺(𝑒)) , 𝑡𝐺(𝑒)), 𝑁𝜃(𝑡𝐹(𝑒),𝑡𝑃(𝑒))≤𝑁𝜃(𝑡𝐺(𝑒),𝑡𝑃(𝑒)), 𝑁𝜃(𝑓𝐹(𝑒),𝑓𝑃(𝑒))≤𝑁𝜃 𝜃 |𝐴∪𝐵| 𝜃 𝑒∈𝐴∩𝐵 (𝑓𝐹(𝑒),𝑓𝐺(𝑒)), 𝑁𝜃(𝑓𝐹(𝑒),𝑓𝑃(𝑒))≤𝑁𝜃(𝑓𝐺(𝑒),𝑓𝑃(𝑒)), and hence 𝑁𝜃 (𝐹(𝑒), 𝑃(𝑒)) ≤𝑁 (𝐹(𝑒), 𝐺(𝑒)) 𝑁 (𝐹(𝑒), 𝑃(𝑒)) ≤𝑁 (𝐺(𝑒), 1 𝜃 , 𝜃 𝜃 𝑁 (𝐹 (𝑒) ,𝐺(𝑒)) = (𝑁 (𝑡 ,𝑡 )+𝑁 (𝑓 ,𝑓 )) . 𝑃(𝑒)) 𝜃 2 𝜃 𝐹(𝑒) 𝐺(𝑒) 𝜃 𝐹(𝑒) 𝐺(𝑒) . Therefore we obtain (7) 1 Proof. (M1) is trivial. 𝑆 ((𝐹, 𝐴) , (𝐺,)) 𝐵 = ∑𝑁 (𝐹 (𝑒) ,𝐺(𝑒)) 𝜃 𝐵 𝜃 (M2) If (𝐹, 𝐴) = (𝐺,,then 𝐵) 𝐴=𝐵and 𝑡𝐹(𝑒) =𝑡𝐺(𝑒), | | 𝑒∈𝐴 𝑓𝐹(𝑒) =𝑓𝐺(𝑒) for any 𝑒∈𝐴.Itfollowsthat𝑁𝜃(𝑡𝐹(𝑒),𝑡𝐺(𝑒))= 1 𝑁 (𝑓 ,𝑓 )=1 𝑁 (𝐹(𝑒), 𝐺(𝑒)) =1 1 , 𝜃 𝐹(𝑒) 𝐺(𝑒) and hence 𝜃 .Thus ≥ ∑𝑁 (𝐹 (𝑒) ,𝑃(𝑒)) 𝑀 ((𝐹, 𝐴), (𝐺, 𝐵)) = (1/|𝐴|) ∑ 𝑁 (𝐹(𝑒), 𝐺(𝑒)) = |𝐴|/|𝐴| = 𝜃 𝜃 𝑒∈𝐴 𝜃 |𝐶| 𝑒∈𝐴 1. Conversely, we assume that 𝑀𝜃((𝐹, 𝐴), (𝐺, 𝐵)).By =1 =𝑆𝜃 ((𝐹, 𝐴) , (𝑃, 𝐶)) , 𝑁𝜃(𝐹(𝑒), 𝐺(𝑒)) ≤1 we have 1=𝑀𝜃((𝐹, 𝐴), (𝐺, 𝐵)) ≤|𝐴∩ 𝐵|/|𝐴 ∪.Itfollowsthat 𝐵| 𝐴=𝐵and 𝑁𝜃(𝑡𝐹(𝑒),𝑡𝐺(𝑒))=1and 1 𝑆 ((𝐺,) 𝐵 , (𝑃, 𝐶)) = ∑𝑁 (𝐺 (𝑒) ,𝑃(𝑒)) 𝑁𝜃(𝑓𝐹(𝑒),𝑓𝐺(𝑒))=1for any 𝑒∈𝐴. If there exist 𝑒∈𝐴,𝑥∈𝑈 𝜃 𝜃 (9) |𝐶| 𝑒∈𝐵 such that 𝑡𝐹(𝑒)(𝑥) =𝑡̸ 𝐺(𝑒)(𝑥),then 󵄨 󵄨 1 ∑ (𝑎 − (𝑎−1) 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨 󵄨 󵄨 ≥ ∑𝑁𝜃 (𝐺 (𝑒) ,𝑃(𝑒)) 𝑥∈𝑈 |𝐶| 𝑒∈𝐴 +𝑏⋅ (𝑡 (𝑥) ,𝑡 (𝑥))) min 𝐹(𝑒) 𝐹(𝑒) 1 ≥ ∑𝑁 (𝐹 (𝑒) ,𝑃(𝑒)) 󵄨 󵄨 |𝐶| 𝜃 = ∑ (𝑎 − 𝑎 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨 𝑒∈𝐴 𝑥∈𝑈

=𝑆𝜃 ((𝐹, 𝐴) , (𝑃, 𝐶)) . +𝑏⋅min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐹(𝑒) (𝑥))) 󵄨 󵄨 + ∑ 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨 𝑥∈𝑈 This completes the proof. 6 The Scientific World Journal

This theorem presents a category of similarity measures Let 𝑎=1, 𝑏=0;then,wehave for vague soft sets. In (7), let 𝑎=0, 𝑏=1;then,wehave 𝑀4 ((𝐹, 𝐴) , (𝐺,)) 𝐵

𝑁𝜃 (𝑡𝐹(𝑒),𝑡𝐺(𝑒)) 1 1 = ∑ (1 − |𝐴∪𝐵| 2 |𝑈| = ∑ min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥)) 𝑒∈𝐴∩𝐵 𝑥∈𝑈 󵄨 󵄨 × ∑ (󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨 󵄨 󵄨 𝑥∈𝑈 ×(∑ ( 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨 𝑥∈𝑈 −1 󵄨 󵄨 + 󵄨𝑓𝐹(𝑒) (𝑥) −𝑓𝐺(𝑒) (𝑥)󵄨)). + min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥))) ) (10) (14) 𝑎=2 𝑏=0 ∑ min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥)) Let , ;then,wehave = 𝑥∈𝑈 , ∑𝑥∈𝑈 max (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥)) 𝑀5 ((𝐹, 𝐴) , (𝐺,)) 𝐵

𝑁𝜃 (𝑓𝐹(𝑒),𝑓𝐺(𝑒)) 󵄨 󵄨 1 ∑𝑥∈𝑈 (1 − 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨) = ∑ ( 󵄨 󵄨 |𝐴∪𝐵| ∑𝑥∈𝑈 (2 − 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨) ∑ min (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥)) 𝑒∈𝐴∩𝐵 = 𝑥∈𝑈 . ∑𝑥∈𝑈 max (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥)) 󵄨 󵄨 ∑𝑥∈𝑈 (1 − 󵄨𝑓𝐹(𝑒) (𝑥) −𝑓𝐺(𝑒) (𝑥)󵄨) + 󵄨 󵄨 ). ∑𝑥∈𝑈 (2 − 󵄨𝑓𝐹(𝑒) (𝑥) −𝑓𝐺(𝑒) (𝑥)󵄨) Therefore we obtain (15)

𝑀1 ((𝐹, 𝐴) , (𝐺,)) 𝐵 Let 𝑎=1, 𝑏=1;then,wehave

1 ∑ min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥)) 𝑀 ((𝐹, 𝐴) , (𝐺,)) 𝐵 = ∑ ( 𝑥∈𝑈 6 2 |𝐴∪𝐵| ∑ max (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥)) 𝑒∈𝐴∩𝐵 𝑥∈𝑈 1 = ∑ min (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥)) 2 |𝐴∪𝐵| + 𝑥∈𝑈 ). ∑ max (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥)) 󵄨 󵄨 𝑥∈𝑈 ∑ 󵄨𝑡 (𝑥) −𝑡 (𝑥)󵄨 (11) × ∑ (2 − 𝑥∈𝑈 󵄨 𝐹(𝑒) 𝐺(𝑒) 󵄨 𝑒∈𝐴∩𝐵 ∑𝑥∈𝑈 (1 + min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥))) 𝑎=0 𝑏=2 󵄨 󵄨 Similarly, let , ;then,wehave ∑ 󵄨𝑓 (𝑥) −𝑓 (𝑥)󵄨 − 𝑥∈𝑈 󵄨 𝐹(𝑒) 𝐺(𝑒) 󵄨 ). ∑𝑥∈𝑈 (1 + min (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥))) 𝑀2 ((𝐹, 𝐴) , (𝐺,)) 𝐵 (16) 1 ∑ min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥)) = ∑ ( 𝑥∈𝑈 Theorem 20. 𝑀 : 𝑉𝑆𝑆(𝑈) × 𝑉𝑆𝑆(𝑈) →[0,1] |𝐴∪𝐵| ∑ (𝑡 (𝑥 )+𝑡 (𝑥 )) 𝛿 is a similarity 𝑒∈𝐴∩𝐵 𝑥∈𝑈 𝐹(𝑒) 𝑖 𝐺(𝑒) 𝑖 measure, where, for any (𝐹, 𝐴), (𝐺, 𝐵) ∈, 𝑉𝑆𝑆(𝑈)

∑ min (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥)) 1 + 𝑥∈𝑈 ). 𝑀 ((𝐹, 𝐴) , (𝐺,)) 𝐵 = ∑ 𝑁 (𝐹 (𝑒) ,𝐺(𝑒)) , 𝛿 |𝐴∪𝐵| 𝛿 ∑𝑥∈𝑈 (𝑓𝐹(𝑒) (𝑥𝑖)+𝑓𝐺(𝑒) (𝑥𝑖)) 𝑒∈𝐴∩𝐵 (12) 1 𝑁𝛿 (𝐹 (𝑒) ,𝐺(𝑒)) = (𝑁𝛿 (𝑡𝐹(𝑒),𝑡𝐺(𝑒))+𝑁𝛿 (𝑓𝐹(𝑒),𝑓𝐺(𝑒))) . 𝑎 = 0.5 𝑏=0 2 Let , ;then,wehave (17)

𝑀3 ((𝐹, 𝐴) , (𝐺,)) 𝐵 Proof. ItcanbeprovedinthesamemannerwithTheorem 19. 󵄨 󵄨 1 ∑ (1 − 󵄨𝑡 (𝑥) −𝑡 (𝑥)󵄨) = ∑ ( 𝑥∈𝑈 󵄨 𝐹(𝑒) 𝐺(𝑒) 󵄨 2 𝐴∪𝐵 󵄨 󵄨 𝑎=0 𝑏=1 | | 𝑒∈𝐴∩𝐵 ∑𝑥∈𝑈 (1 + 󵄨𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)󵄨) In (17), let , ;then,wehave 󵄨 󵄨 𝑀 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ∑𝑥∈𝑈 (1 − 󵄨𝑓𝐹(𝑒) (𝑥) −𝑓𝐺(𝑒) (𝑥)󵄨) 7 + 󵄨 󵄨 ). ∑ (1 + 󵄨𝑓 (𝑥) −𝑓 (𝑥)󵄨) 1 𝑥∈𝑈 󵄨 𝐹(𝑒) 𝐺(𝑒) 󵄨 = (13) 2 |𝐴∪𝐵| The Scientific World Journal 7

∑ (1 − max (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥))) × ∑ ( 𝑥∈𝑈 Similarly, we have 𝑒∈𝐴∩𝐵 ∑𝑥∈𝑈 (1 − min (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥))) 𝑀2 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ≈ 0.4,

∑𝑥∈𝑈 (1 − max (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥))) + ). 𝑀3 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ≈ 0.39, ∑𝑥∈𝑈 (1 − min (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥))) (18) 𝑀4 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ≈ 0.44, 𝑎=0 𝑏=2 Let , ;then,wehave 𝑀5 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ≈ 0.47, (23) 𝑀8 ((𝐹, 𝐴) , (𝐺,)) 𝐵 𝑀6 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ≈ 0.45, 1 = |𝐴∪𝐵| 𝑀7 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ≈ 0.41, 𝑀 ((𝐹, 𝐴) , (𝐺,)) 𝐵 ≈ 0.45. ∑ (1 − max (𝑡𝐹(𝑒) (𝑥) ,𝑡𝐺(𝑒) (𝑥))) 8 × ∑ ( 𝑥∈𝑈 ∑𝑥∈𝑈 (2 − 𝑡𝐹(𝑒) (𝑥) −𝑡𝐺(𝑒) (𝑥)) 𝑒∈𝐴∩𝐵 Let 𝐴, 𝐵 ∈ 𝐹(𝑈).Weconsider𝑁𝜃(𝐴, 𝐵).Forany𝑥∈𝑈,by 1−|𝐴(𝑥)−𝐵(𝑥)|,min ≥0 (𝐴(𝑥), 𝐵(𝑥)), ≥0 𝑎−𝑎|𝐴(𝑥)−𝐵(𝑥)|+ ∑ (1 − max (𝑓𝐹(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥))) + 𝑥∈𝑈 ). 𝑏⋅min(𝐴(𝑥), 𝐵(𝑥)) = 𝑎(1−|𝐴(𝑥)−𝐵(𝑥)|)+𝑏⋅min(𝐴(𝑥), 𝐵(𝑥)) ∑ (2 − 𝑓 (𝑥) −𝑓 (𝑥)) 𝑥∈𝑈 𝐹(𝑒) 𝐺(𝑒) we conclude that ∑𝑥∈𝑈(𝑎−𝑎|𝐴(𝑥)−𝐵(𝑥)|+𝑏⋅min(𝐴(𝑥), 𝐵(𝑥))) (19) is increasing with respect to 𝑎 and 𝑏. Therefore 𝑈={𝑥,𝑥 } 𝐸={𝑒,𝑒 ,𝑒 ,𝑒 } 𝐴={𝑒,𝑒 , Example 21. Let 1 2 , 1 2 3 4 , 1 2 ∑𝑥∈𝑈 |𝐴 (𝑥) −𝐵(𝑥)| 𝑒3},and𝐵={𝑒1,𝑒2,𝑒4}.Supposethat(𝐹, 𝐴) and (𝐺, 𝐵) are (24) ∑ (𝑎−𝑎|𝐴 (𝑥) −𝐵(𝑥)| +𝑏⋅min (𝐴 (𝑥) ,𝐵(𝑥))) vague soft sets over 𝑈 given by 𝑥∈𝑈 [0, 0.5] [0.2, 0.4] is decreasing with respect to 𝑎 and 𝑏.By 𝐹(𝑒1)= + , 𝑥1 𝑥2 𝑁𝜃 (𝐴,) 𝐵 [0.3, 0.6] [0.4, 0.5] 𝐺(𝑒1)= + , 𝑥1 𝑥2 =1(1+(∑ |𝐴 (𝑥) −𝐵(𝑥)| [0.5, 0.5] [0.5, 0.5] 𝑥∈𝑈 𝐹(𝑒2)= + , 𝑥1 𝑥2 × ∑ (𝑎−𝑎|𝐴 (𝑥) −𝐵(𝑥)| (25) (20) 𝑥∈𝑈 [0.5, 0.5] [0.4, 0.7] 𝐺(𝑒2)= + , −1 −1 𝑥1 𝑥2 +𝑏 ⋅ min (𝐴 (𝑥) ,𝐵(𝑥))) ) ) , [0, 0] [1, 1] 𝐹(𝑒3)= + , 𝑥1 𝑥2 it follows that 𝑁𝜃(𝐴, 𝐵) is increasing with respect to 𝑎 and 𝑏. [0, 0] [1, 1] Thus we have the following corollary. 𝐺(𝑒4)= + . 𝑥1 𝑥2 Corollary 22. Let (𝐹, 𝐴), (𝐺, 𝐵) ∈.Then 𝑉𝑆𝑆(𝑈) By the definition, we have (1) 𝑀1((𝐹, 𝐴), (𝐺, 𝐵))2 ≤𝑀 ((𝐹, 𝐴), (𝐺, 𝐵)),1 𝑀 ((𝐹, 𝐴), ∑𝑥∈𝑈 min (𝑡𝐹(𝑒 ) (𝑥) ,𝑡𝐺(𝑒 ) (𝑥)) 0 + 0.2 2 (𝐺, 𝐵)) ≤𝑀 ((𝐹, 𝐴), (𝐺, 𝐵)) 1 1 = = , 6 ; ∑ (𝑡 (𝑥) ,𝑡 (𝑥)) 0.3 + 0.4 7 𝑀 ((𝐹, 𝐴), (𝐺, 𝐵)) ≤𝑀 ((𝐹, 𝐴), (𝐺, 𝐵)) ≤𝑀 ((𝐹, 𝐴), 𝑥∈𝑈 max 𝐹(𝑒1) 𝐺(𝑒1) (2) 3 4 5 (𝐺, 𝐵)); ∑𝑥∈𝑈 min (𝑓𝐹(𝑒 ) (𝑥) ,𝑓𝐺(𝑒 ) (𝑥)) 0.4 + 0.5 9 1 1 = = , (3) (𝑀4((𝐹, 𝐴), (𝐺, 𝐵))6 ≤𝑀 ((𝐹, 𝐴), (𝐺,. 𝐵)) ∑ (𝑓 (𝑥) ,𝑓 (𝑥)) 0.5 + 0.6 11 𝑥∈𝑈 max 𝐹(𝑒1) 𝐺(𝑒1) (21) This corollary shows the relationships among the similar- ∑𝑥∈𝑈 min (𝑡𝐹(𝑒 ) (𝑥) ,𝑡𝐺(𝑒 ) (𝑥)) 0.5 + 0.4 9 ity measures presented in this section. Similarly, we have the 2 2 = = , following. ∑ (𝑡 (𝑥) ,𝑡 (𝑥)) 0.5 + 0.5 10 𝑥∈𝑈 max 𝐹(𝑒2) 𝐺(𝑒2) Corollary 23. Let (𝐹, 𝐴), (𝐺, 𝐵) ∈.Then 𝑉𝑆𝑆(𝑈) 𝑀7((𝐹, 𝐴), ∑ (𝑓 (𝑥) ,𝑓 (𝑥)) 𝑥∈𝑈 min 𝐹(𝑒 ) 𝐺(𝑒 ) 0.5 + 0.3 8 (𝐺, 𝐵))8 ≤𝑀 ((𝐹, 𝐴), (𝐺,. 𝐵)) 2 2 = = . ∑ (𝑓 (𝑥) ,𝑓 (𝑥)) 0.5 + 0.5 10 𝑥∈𝑈 max 𝐹(𝑒2) 𝐺(𝑒2) 5. Entropy Measures for Fuzzy Soft Sets It follows that 1 2 9 9 8 In this section, we present some entropies for vague soft sets 𝑀 ((𝐹, 𝐴) , (𝐺,)) 𝐵 = ( + + + ) ≈ 0.35. 1 8 7 11 10 10 (22) in the sense of Definition 13. 8 The Scientific World Journal

ℎ(𝑦 ,𝑧 ,...,𝑦 ,𝑧 ) Theorem 24. 𝐻𝜃(𝐹, 𝐴) is an entropy, where 1 1 𝑛 𝑛 𝑛 𝑐 󵄨 󵄨 𝐻𝜃 (𝐹, 𝐴) =𝑀𝜃 ((𝐹, 𝐴) ,(𝐹 ,𝐴)) (26) = ∑ (𝑎 − (𝑎−1) 󵄨𝑦𝑖 −𝑧𝑖󵄨 +𝑏⋅min (𝑦𝑖,𝑧𝑖)) , 𝑖=1 (𝐹, 𝐴) ∈ 𝑉𝑆𝑆(𝑈) for any . 𝑔(𝑦1,𝑧1,...,𝑦𝑛,𝑧𝑛) 𝑓(𝑦1,𝑧1,...,𝑦𝑛,𝑧𝑛)= , ℎ(𝑦1,𝑧1,...,𝑦𝑛,𝑧𝑛) Proof. (H1) We note that 𝑀𝜃 is a similarity measure. For any (𝐹, 𝐴) ∈ VSS(𝑈),wehave𝐻𝜃(𝐹, 𝐴) = 0𝜃 ⇔𝑀 ((𝐹, 𝐴), (28) 𝑐 (𝐹 , 𝐴)) = 0 ⇔ for all 𝑒∈𝐴, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =, 0 𝑓𝐹(𝑒)(𝑥) =, 1 𝑡𝐹𝑐(𝑒)(𝑥) =, 1 𝑓𝐹𝐶(𝑒)(𝑥) =,or 0 𝑡𝐹(𝑒)(𝑥) =, 1 𝑓𝐹(𝑒)(𝑥) =, 0 where 𝑦𝑖,𝑧𝑖 ∈ [0, 1], 𝑦𝑖 +𝑧𝑖 ≤1.If𝑦𝑗 ≤𝑧𝑗,thenwehave 𝑡𝐹𝑐(𝑒)(𝑥) =, 0 𝑓𝐹𝑐(𝑒)(𝑥) = 1 ⇔ for all 𝑒∈𝐴, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =, 0 𝑓𝐹(𝑒)(𝑥) =,or 1 𝑡𝐹(𝑒)(𝑥) =, 1 𝑓𝐹(𝑒)(𝑥) =. 0 𝜕𝑓 (H2) For any (𝐹, 𝐴) ∈ VSS(𝑈),wehave𝐻𝜃(𝐹, 𝐴) = 1⇔ 𝑐 𝑐 𝜕𝑦𝑗 𝑀𝜃((𝐹, 𝐴), (𝐹 ,𝐴))= 1 ⇔ (𝐹, 𝐴) = (𝐹 ,𝐴)⇔ for all 𝑒∈𝐴, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =𝐹 𝑡 𝑐(𝑒)(𝑥), 𝑓𝐹(𝑒)(𝑥) =𝐹 𝑓 𝑐(𝑒)(𝑥) ⇔ for all 𝑒∈ 𝑛 󵄨 󵄨 𝑛 𝑏∑𝑖=1 󵄨𝑦𝑖 −𝑧𝑖󵄨 +∑𝑖=1 (𝑎 + 𝑏 min (𝑦𝑖,𝑧𝑖)) 𝐴, 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) =𝐹(𝑒) 𝑓 (𝑥). = ≥0, ℎ2 (𝑦 ,𝑧 ,...,𝑦 ,𝑧 ) (H3) is trivial. 1 1 𝑛 𝑛 (29) (H4) Let (𝐹, 𝐴), (𝐺, 𝐴)∈ VSS(𝑈),andforall𝑒∈𝐴, 𝜕𝑓 𝑥∈𝑈, 𝑡𝐹(𝑒)(𝑥) ≤𝐺(𝑒) 𝑡 (𝑥), 𝑓𝐹(𝑒)(𝑥) ≥𝐺(𝑒) 𝑓 (𝑥) if 𝑡𝐺(𝑒)(𝑥) ≤ 𝜕𝑧𝑗 𝑓𝐺(𝑒)(𝑥),and𝑡𝐹(𝑒)(𝑥) ≥𝐺(𝑒) 𝑡 (𝑥), 𝑓𝐹(𝑒)(𝑥) ≤𝐺(𝑒) 𝑓 (𝑥) if 𝑡𝐺(𝑒)(𝑥) ≥𝑓𝐺(𝑒)(𝑥).Wenotethat𝑁𝜃(𝑡𝐹(𝑒),𝑡𝐹𝑐(𝑒))=𝑁𝜃(𝑓𝐹(𝑒),𝑓𝐹𝑐(𝑒)) 𝑛 󵄨 󵄨 (−𝑎) ∑𝑖=1 󵄨𝑦𝑖 −𝑧𝑖󵄨 −𝑔(𝑦1,𝑧1,...,𝑦𝑛,𝑧𝑛) =𝑁𝜃(𝑡𝐹(𝑒),𝑓𝐹(𝑒)) and hence 𝐻𝜃(𝐹, 𝐴) = (1/|𝐴|) ∑ 𝑁𝜃 = ≤0. 𝑒∈𝐴 ℎ2 (𝑦 ,𝑧 ,...,𝑦 ,𝑧 ) (𝑡𝐹(𝑒),𝑓𝐹(𝑒)). 1 1 𝑛 𝑛 By the definition, for each 𝑒∈𝐴,wehave Therefore, we can conclude that 𝑓 is increasing with respect to 𝑦𝑗 and decreasing with respect to 𝑧𝑗 if 𝑦𝑗 ≤𝑧𝑗.Similarly,𝑓 𝑁𝜃 (𝑡𝐹(𝑒),𝑓𝐹(𝑒)) is decreasing with respect to 𝑦𝑗 and increasing with respect to 󵄨 󵄨 󵄨 󵄨 𝑧𝑗 if 𝑦𝑗 ≥𝑧𝑗.Itfollowsthat𝑁𝜃(𝑡𝐹(𝑒),𝑓𝐹(𝑒))≤𝑁𝜃(𝑡𝐺(𝑒),𝑓𝐺(𝑒)) = ∑ (𝑎 − 𝑎 󵄨𝑡𝐹(𝑒) (𝑥) −𝑓𝐹(𝑒) (𝑥)󵄨 𝑥∈𝑈 and consequently, 𝐻𝜃(𝐹, 𝐴)𝜃 ≤𝐻 (𝐺, 𝐴).

+𝑏⋅min (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥))) Theorem 25. 𝐻𝛿(𝐹, 𝐴) is an entropy, where

󵄨 󵄨 𝑐 󵄨 󵄨 𝐻𝛿 (𝐹, 𝐴) =𝑀𝛿 ((𝐹, 𝐴) ,(𝐹 ,𝐴)) (30) ×(∑ (𝑎 − (𝑎−1) 󵄨𝑡𝐹(𝑒) (𝑥) −𝑓𝐹(𝑒) (𝑥)󵄨 𝑥∈𝑈

−1 for any (𝐹, 𝐴) ∈ 𝑉𝑆𝑆(𝑈). +𝑏⋅ (𝑡 (𝑥) ,𝑓 (𝑥))) ) , min 𝐹(𝑒) 𝐹(𝑒) Proof. The proof is similar to that of Theorem 24. (27) 𝑁𝜃 (𝑡𝐺(𝑒),𝑓𝐺(𝑒)) Using similarity measures 𝑀𝑖 (1≤𝑖≤8),wecanobtain 󵄨 󵄨 𝐻 (1≤𝑖≤8) = ∑ (𝑎 − 𝑎 󵄨𝑡𝐺(𝑒) (𝑥) −𝑓𝐺(𝑒) (𝑥)󵄨 the corresponding entropies 𝑖 as follows: 𝑥∈𝑈

1 ∑𝑥∈𝑈 min (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥)) +𝑏⋅min (𝑡𝐺(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥))) 𝐻 (𝐹, 𝐴) = ∑ , 1 𝐴 | | 𝑒∈𝐴 ∑𝑥∈𝑈 max (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥)) 󵄨 󵄨 ×(∑ (𝑎 − (𝑎−1) 󵄨𝑡𝐺(𝑒) (𝑥) −𝑓𝐺(𝑒) (𝑥)󵄨 󵄨 󵄨 2 ∑𝑥∈𝑈 min (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥)) 𝑥∈𝑈 𝐻2 (𝐹, 𝐴) = ∑ , |𝐴| ∑𝑥∈𝑈 (𝑡𝐹(𝑒) (𝑥) +𝑓𝐹(𝑒) (𝑥)) −1 𝑒∈𝐴 󵄨 󵄨 +𝑏⋅min (𝑡𝐺(𝑒) (𝑥) ,𝑓𝐺(𝑒) (𝑥))) ) . 1 ∑ (1 − 󵄨𝑡 (𝑥) −𝑓 (𝑥)󵄨) 𝐻 (𝐹, 𝐴) = ∑ 𝑥∈𝑈 󵄨 𝐹(𝑒) 𝐹(𝑒) 󵄨 , 3 𝐴 󵄨 󵄨 | | 𝑒∈𝐴 ∑𝑥∈𝑈 (1 + 󵄨𝑡𝐹(𝑒) (𝑥) −𝑓𝐹(𝑒) (𝑥)󵄨)

Let 𝑈={𝑥1,𝑥2,...,𝑥𝑛}. Considering the following functions: 1 1 󵄨 󵄨 𝐻 (𝐹, 𝐴) = ∑ (1 − ∑ 󵄨𝑡 (𝑥) −𝑓 (𝑥)󵄨), 4 |𝐴| |𝑈| 󵄨 𝐹(𝑒) 𝐹(𝑒) 󵄨 𝑔(𝑦1,𝑧1,...,𝑦𝑛,𝑧𝑛) 𝑒∈𝐴 𝑥∈𝑈 𝑛 󵄨 󵄨 󵄨 󵄨 2 ∑𝑥∈𝑈 (1 − 󵄨𝑡𝐹(𝑒) (𝑥) −𝑓𝐹(𝑒) (𝑥)󵄨) = ∑ (𝑎 − 𝑎 󵄨𝑦 −𝑧󵄨 +𝑏⋅ (𝑦 ,𝑧)) , 𝐻 (𝐹, 𝐴) = ∑ 󵄨 󵄨 , 󵄨 𝑖 𝑖󵄨 min 𝑖 𝑖 5 |𝐴| ∑ (2 − 󵄨𝑡 𝑥 −𝑓 𝑥 󵄨) 𝑖=1 𝑒∈𝐴 𝑥∈𝑈 󵄨 𝐹(𝑒) ( ) 𝐹(𝑒) ( )󵄨 The Scientific World Journal 9

𝐻 (𝐹, 𝐴) 6 󵄨 󵄨 1 ∑ 󵄨𝑡 (𝑥) −𝑓 (𝑥)󵄨 = ∑ (1 − 𝑥∈𝑈 󵄨 𝐹(𝑒) 𝐹(𝑒) 󵄨 ), [1] D. Molodtsov, “Soft set theory—first results,” Computers & 𝐴 Mathematics with Applications,vol.37,no.4-5,pp.19–31,1999. | | 𝑒∈𝐴 ∑𝑥∈𝑈 (1 + min (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥))) [2] N. C¸agman˘ and S. Enginoglu,˘ “Soft set theory and uni-int 1 ∑ (1 − max (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥))) decision making,” European Journal of Operational Research, 𝐻 (𝐹, 𝐴) = ∑ 𝑥∈𝑈 , 7 𝐴 vol. 207, no. 2, pp. 848–855, 2010. | | 𝑒∈𝐴 ∑𝑥∈𝑈 (1 − min (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥))) [3] F. Feng, Y. Li, and N. C¸agman,˘ “Generalized uni–int decision making schemes based on choice value soft sets,” European 2 ∑𝑥∈𝑈 (1 − max (𝑡𝐹(𝑒) (𝑥) ,𝑓𝐹(𝑒) (𝑥))) 𝐻 (𝐹, 𝐴) = . Journal of Operational Research,vol.220,no.1,pp.162–170,2012. 8 |𝐴| ∑ (2 − 𝑡 (𝑥) −𝑓 (𝑥)) 𝑥∈𝑈 𝐹(𝑒) 𝐹(𝑒) [4] F. Feng and Y. Li, “Soft subsets and soft product operations,” (31) Information Sciences,vol.232,pp.44–57,2013. [5]S.J.KalayathankalandG.SureshSingh,“Afuzzysoftflood Example 26. We consider the vague soft set (𝐹, 𝐴) given in alarm model,” Mathematics and Computers in Simulation,vol. Example 21 80,no.5,pp.887–893,2010. [6]P.K.Maji,A.R.Roy,andR.Biswas,“Anapplicationofsoftsets 1 0 + 0.2 0.5 + 0.5 0+0 in a decision making problem,” Computers & Mathematics with 𝐻 (𝐹, 𝐴) = ( + + ) ≈ 0.39, Applications,vol.44,no.8-9,pp.1077–1083,2002. 1 3 0.5 + 0.6 0.5 + 0.5 1+1 [7] D. Molodtsov, The Theory of Soft Sets,URSSPublishers, 𝐻2 (𝐹, 𝐴) ≈ 0.44, 𝐻3 (𝐹, 𝐴) ≈ 0.46, Moscow, Russia, 2004 (Russian). [8]M.M.Mushrif,S.Sengupta,andA.K.Roy,“Textureclas- 𝐻4 (𝐹, 𝐴) ≈ 0.52, 𝐻5 (𝐹, 𝐴) ≈ 0.57, sification using a novel, soft-set theory based classification 𝐻 (𝐹, 𝐴) ≈ 0.53, 𝐻 (𝐹, 𝐴) = 0.5, 𝐻 (𝐹, 𝐴) ≈ 0.56. algorithm,” in Computer Vision—ACCV 2006,vol.3851of 6 7 8 Lecture Notes in Computer Science, pp. 246–254, 2006. (32) [9] A. R. Roy and P. K. Maji, “A fuzzy soft set theoretic approach to decision making problems,” Journal of Computational and 6. Concluding Remarks Applied Mathematics,vol.203,no.2,pp.412–418,2007. [10] Z.Xiao,K.Gong,andY.Zou,“Acombinedforecastingapproach Soft set theory was originally proposed as a general math- based on fuzzy soft sets,” Journal of Computational and Applied ematical tool for dealing with uncertainties. Wang and Qu Mathematics,vol.228,no.1,pp.326–333,2009. [32] introduced axiomatic definitions of entropy, similarity [11] Y. Zou and Z. Xiao, “Data analysis approaches of soft sets under measure, and distance measure for vague soft sets and incomplete information,” Knowledge-Based Systems,vol.21,no. proposed some formulas to calculate them. This paper is 8, pp. 941–945, 2008. devoted to a further discussion along this line. We point [12] P.K.Maji,R.Biswas,andA.R.Roy,“Softsettheory,”Computers out that there are some drawbacks in [32] by examples. We & Mathematics with Applications,vol.45,no.4-5,pp.555–562, propose a new axiomatic definition of entropy and present 2003. a new approach to construct the similarity measures and [13]M.I.Ali,F.Feng,X.Liu,W...Min,andM.Shabir,“Onsome entropies for vague soft sets. Based on these uncertainty new operations in soft set theory,” Computers & Mathematics measures, we can further probe the applications of vague soft with Applications,vol.57,no.9,pp.1547–1553,2009. sets in the fields such as pattern recognition, data analysis, [14] K.-Y. Qin and Z.-Y. Hong, “On soft equality,” Journal of anddecisionmaking. Computational and Applied Mathematics,vol.234,no.5,pp. 1347–1355, 2010. [15] P.K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” The Journal Conflict of Interests of Fuzzy Mathematics,vol.9,no.3,pp.589–602,2001. The authors declare that they have no financial and personal [16] P. Majumdar and S. K. Samanta, “Generalised fuzzy soft sets,” Computers & Mathematics with Applications,vol.59,no.4,pp. relationships with other people or organizations that can 1425–1432, 2010. inappropriately influence the paper. There are no professional or other personal interests of any nature or kind in any [17] P. K. Maji, R. Biswas, and A. R. Roy, “Intuitionistic fuzzy soft sets,” Journal of Fuzzy Mathematics,vol.9,no.3,pp.677–692, product, service, and/or company that could be construed as 2001. influencing the position presented in, or the review of, the paper entitled. [18] P.K. Maji, A. R. Roy, and R. Biswas, “On intuitionistic fuzzy soft sets,” The Journal of Fuzzy Mathematics,vol.12,no.3,pp.669– 683, 2004. Acknowledgments [19]X.B.Yang,T.Y.Lin,J.Y.Y.Yang,Y.Li,andD.J.Yu,“Combi- nation of interval-valued fuzzy set and soft set,” Computers & This work has been partially supported by the National Mathematics with Applications,vol.58,no.3,pp.521–527,2009. Natural Science Foundation of China (Grant nos. 61372187, [20] W. Xu, J. Ma, S. Wang, and G. Hao, “Vague soft sets and their 61175044) and The Program of Education Office of Sichuan properties,” Computers & Mathematics with Applications,vol. Province (Grant no. 11ZB068). 59,no.2,pp.787–794,2010. 10 The Scientific World Journal

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