#### Abstract

Molodtsov has introduced the concept of soft sets and the application of soft sets in decision making and medical diagnosis problems. The basic properties of vague soft sets are presented. In this paper, we introduce the concept of interval-valued vague soft sets which are an extension of the soft set and its operations such as equality, subset, intersection, union, AND operation, OR operation, complement, and null while further studying some properties. We give examples for these concepts, and we give a number of applications on interval-valued vague soft sets.

#### 1. Introduction

Uncertain or imprecise data are inherent and pervasive in many important applications in areas such as economics, engineering, environmental sciences, social science, medical science, and business management. There have been a number of researches and applications in the literature dealing with uncertainties such as Molodtsov’s [1] who initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties which is free, fuzzy set theory by Zadeh [2], rough set theory by Pawlak [3], vague set theory by Gau and Buehrer [4], intuitionistic fuzzy set theory by Atanassov [5], and interval mathematics by Atanassov [6]. Alkhazaleh et al. [7] also introduced the concept of fuzzy parameterized interval-valued fuzzy soft sets and established its application in decision making. Alkhazaleh et al. [8] further introduced soft multisets as a generalization of Molodtsov's soft set. Bustince and Burillo [9] studied the difference between vague soft sets and intuitionistic fuzzy soft sets. Furthermore, soft sets, soft groups, and new operation in soft set theory were studied by Aktas and Cagman [10], Ali et al. [11], Maji et al. [12–14], Jiang et al. [15], and Alhazaymeh et al. [16].

The purpose of this paper is to further extend the concept of vague set theory by introducing the notion of a vague soft set and deriving its basic properties. The paper is organized as follows. Section 1 is the introduction followed by Section 2 which is preliminaries of vague set and vague soft set. Section 3 presents the basic concepts and definitions for a vague soft set and a vague set and then redefines the concept of the intersection of two soft sets. Section 4 introduces the notion of an interval-valued vague soft set and discusses its properties. Concluding remarks and open questions for further investigation are provided in Section 5.

#### 2. Preliminaries

A soft set is a mapping from a set of parameters to the power set of a universe set. However, the notion of a soft set, as given in its definition, cannot be used to represent the vagueness of the associated parameters. In this section, we provide the concept of a vague soft set based on soft set theory and vague set theory and the basic properties.

Let be a universe, a set of parameters, the power set of the vague sets on , and .

*Definition 1 (see [1]). *A pair is called a soft set over , where is a mapping given by .

In other words, a soft set over is a parameterized family of subset of the universe .

*Definition 2 (see [17]). * A pair is called a vague soft set over , where is a mapping given by .

In other words, a vague soft set over is a parameterized family of the universe . For , is regarded as the set of -approximate of the vague soft set .

*Definition 3 (see [17]). *For two vague soft sets and over universe , we say that is the vague subset of , if , and are identical approximations. This relationship is denoted by .

*Definition 4 (see [17]). * Two vague soft sets and over universe are said to be vague soft equal if is a vague soft subset of and is a vague soft subset of .

*Definition 5 (see [17]). *The complement of vague soft set is denoted by and is defined by , where is a mapping given by , , .

*Definition 6 (see [17]). * A vague soft set over is said to be a null vague soft set denoted by , , and , .

*Definition 7 (see [17]). * A vague soft set over is said to be an absolute vague soft set denoted by , , and , .

*Definition 8 (see [17]). *If and are two vague soft sets over . “ and ” denoted by “” which is defined by , where , , .

*Definition 9 (see [17]). * If and are two vague soft sets over . “ or ” denoted by “” is defined by , where , and , .

*Definition 10 (see [18]). * An interval-valued fuzzy set is a mapping such that
where stands for the set of all closed subintervals of , the set of all interval-valued fuzzy sets on is denoted by .

The complement, intersection, and union of the interval-valued fuzzy sets are defined in [19] as follows. Let then(1) the complement of is denoted by , where (2)the intersection of and is denoted by , where (3)the union of and is denoted by , where

We used these definitions to introduce the concept of interval-valued vague soft set. Also, we extend these definitions to provide some basic operation on interval-valued vague soft set, such as equality, subset, intersection, union, AND operation, OR operation, complement, and null.

#### 3. Interval-Valued Vague Soft Set

In this section, we introduce the state of interval-valued vague soft set and some operations. These are equality, subset, intersection, union, AND operation, OR operation, complement, and null.

Let be an initial universe, a set of parameters, the power set of interval-valued vague sets on , and . The concept of an interval-valued vague soft set is given by the following proposed definition.

*Definition 11. *A pair is called an interval-valued vague soft set over , where is a mapping given by .

In other words, an interval-valued vague soft set over is a parameterized family of an interval-valued vague set of the universe .

*Example 12. *Consider an interval-valued vague soft set , where is the set of three cars under the consideration of the decision maker for purchase, which is denoted by , and are the parameters set, where sporty, family, utility}. The interval-valued vague soft set describes the “attractiveness of the cars" to this decision maker.

Suppose that

The interval-valued vague soft set is a parameterized family of interval-valued vague soft set on , and sporty cars , , , family cars , , , utility cars , , .

*Definition 13. *For two interval-valued vague soft sets and over universe , we say that is an interval-valued vague soft subset of , if and , and are identical approximations. This relationship is denoted by . Similarly, is said to be interval-valued vague soft superset of and is an interval-valued vague soft subset of as follows denoted by . Consider the definition of vague subsets. Let and be two vague sets of the universe . If , and , then the vague set are included by , denoted by , where .

*Definition 14. *Two interval-valued vague soft sets and over universe are said to be interval-valued vague soft equal if is an interval-valued vague soft subset of and is an interval-valued vague soft subset of .

*Definition 15. *Let be a parameter set. The not set of denoted by is defined by , where = not .

*Definition 16. *The complement of an interval-valued vague soft set is denoted by and is defined by , where is a mapping given by , , and , .

*Example 17. *We used the descriptions from Example 12 to illustrate the complement of interval-valued vague soft set as not sporty cars , , , not family cars = , , , not utility cars , , .

*Definition 18. *An interval-valued vague soft set over is said to be a null interval-valued vague soft set denoted by , if , , , and , .

*Definition 19. * An interval-valued vague soft set over is said to be an absolute interval-valued vague soft set denoted by , if , , , and , .

*Definition 20. *If and are two interval-valued vague soft sets over , “ and ” is an interval-valued vague soft set denoted by “” which is defined by , where , , that is .

*Definition 21. *If and are two interval-valued vague soft sets over , “” is an interval-valued vague soft set denoted by “” which is defined by , where , , that is , , .

*Definition 22. * The union of two interval-valued vague soft sets and over a universe is an interval-valued vague soft set , where and ,Hence, .

*Definition 23. *The intersection of two interval-valued vague soft sets and over a universe is an interval-valued vague soft set , where and , Hence, .

Theorem 24. *If and are two interval-valued vague soft sets over , then one has the following properties:*(i)*;*(ii)*.*

*Proof. * Assume that , where and Since , then we have , where for all and . Hence,Since and , then we have .

Suppose that , where and we take Therefore, and are the same operators. Thus, ;

(ii) the proof is similar to that of (i).

#### 4. An Application on Interval-Valued Vague Soft Set

In this section, we provide an application of interval-valued vague soft set.

Let be the set of apartments having different furnishings and rental, with the parameters set, fully furnished, partially furnished, empty, monthly, yearly, weekly}. Let and denote two subsets of the set of parameters . Also let represent the furnished fully furnished, partially furnished, empty} and represent the rental = {monthly, yearly, weekly}.

Assuming that an interval-valued vague soft set describes the “apartments having furnished,” an interval-valued vague soft set describes the “apartments having rental.” These interval-valued vague soft sets may be computed as below.

An interval-valued vague soft set is defined as apartments being fully furnished = , , , partially furnished = , , , empty = , , .

An interval-valued vague soft set is defined as apartments having monthly rental = , , , apartments having yearly rental = , , , apartments having weekly rental = , , .

Let and be a two interval-valued vague soft sets over the common universe . After performing some operations (such as AND and OR) on an interval-valued vague soft set for some particular parameters of and , we obtain another interval-valued vague soft set. The newly obtained interval-valued vague soft set is termed as a resultant interval-valued vague soft set of and .

Suppose that Mr. X is interested to rent an apartment on the basis of his choice parameters, which constitute the subset fully furnished, monthly, partially furnished} of the set , and = {partially furnished, security, yearly}, where both and . This means that out of available apartments in , he is to select an apartment which qualify all parameters of an interval-value vague soft sets and . The problem is to select the apartment which is most suitable with the choice parameters of Mr. X.

To solve this problem, we require some concepts in the soft set theory of Molodtsov [1], which are presented below.

Consider the above two interval-valued vague soft sets and as in Tables 1 and 2, respectively. If we perform “ AND ,” then we will have parameters of the form , . If we require the interval-valued vague soft set for all the parameters , then the resultant interval-valued vague soft set for the interval-valued vague soft sets and will be .

As such, after performing the “” for some parameters, the tabular representation of truth membership of the interval-valued vague soft set will likely take a form as in Table 3.

Representation of the false-membership function of and are shown in Tables 4 and 5.

After performing the “ AND ” for parameters tabular representation of false-membership of the interval-valued vague soft set will take a form as in Table 6.

Representations of the comparison for truth-membership function and false-membership function are shown in Tables 7 and 8, respectively.

Tables 9 and 10 show the calculated truth-membership score and false-membership score, respectively, while Table 11 shows the final score.

Clearly the maximum score is , which is the score of the apartment . Mr. X's best option is to rent apartment , while his second best choice will be .

#### 5. Conclusion

In this paper, the basic concept of a soft set is reviewed. We introduce the notion of an interval-valued vague soft set as an extension to the vague soft set. The basic properties of interval-valued vague soft sets are also presented. These are complement, null, union, intersection, quality, subsets, “AND” and “OR” operators, and the application with respect to the interval-valued vague soft set is illustrated.

It is desirable to further explore the applications of using the interval-valued vague soft set approach to problems such as decision making, forecasting, and data analysis.

#### Acknowledgments

The authors are indebted to Universiti Kebangsaan Malaysia for funding this research under the Grant of UKM-GUP-2011-159.