Decomposition theorems and representation theorems of vague soft sets

Abstract

A vague soft set is a combination of a vague set and a soft set. In this paper, four kinds of cut sets on vague soft sets are introduced, which are generations of cut sets on fuzzy sets and have the same properties as that of fuzzy sets. The relations among these kinds of cut sets are also discussed. Then, the definitions of order nested sets and inverse order nested sets are given. Furthermore, based on them, the decomposition theorems and representation theorems of vague soft sets are established.

Keywords

Vague soft set cut set nested sets decomposition theorems representation theorems

1 Introduction

Since the initation of fuzzy sets introduced by Zadeh in [1], the theories of fuzzy sets and fuzzy systems are developed rapidly. The cut set is the bridge connecting the fuzzy sets and classical sets. Based on the cut sets, the decomposition theorems and representation theorems can be established [2]. Yuan et al. [3] introduced three new cut sets on fuzzy sets by using the neighborhood relations between fuzzy point and fuzzy set and also pointed out four kinds of cut sets have similar properties. Yuan et al. [4] introduced the definitions of cut sets on intuitionistic fuzzy sets and interval valued fuzzy sets by triple valued fuzzy sets, which breaks through the restriction that the cut sets are classical sets. They also established the decomposition theorems and representation theorems for each kind of cut set of intuitionistic fuzzy sets and interval valued fuzzy sets. Wang et al. [5] proved the equivalence of the four decomposition theorems and the equivalence of the four representation theorems obtained in [4]. Li et al. [6] proposed four kinds of cut sets on three-demensional fuzzy sets which is a new kind of L-fuzzy set, and established the decomposition theorems and representation theorems.

With the development of the theory on fuzzy sets, intuitionistic fuzzy sets [7], vague sets [8], interval mathematics [9] and other mathematical tools are introduced to describe uncertainty. However, all of these theories have their own difficulties which have been pointed out in [10]. Molodtsov suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tools of these theories. To overcome these difficulties, Molodtsov [10] introduced the concept of soft sets as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Maji et al. [11] introduced the concept of fuzzy soft sets by combining fuzzy sets and soft sets. Xu et al. [12] introduced the notion of vague soft sets, derived its basic properties and illustrated its potential applications.

Since vague sets are equivalent to intuitionistic fuzzy sets [13], so vague soft sets are equivalent to intuitionistic fuzzy soft sets [14, 15]. Some scholars have studied intuitionistic fuzzy soft sets from different aspects. For example, Gunduz and Bayramov [16] introduced the concept of an intuitionistic fuzzy soft module and some operations on intuitionistic fuzzy soft sets were given, they also studied some of its basic properties. Jiang et al. [17] proposed the notion of the interval-valued intuitionisitc fuzzy soft set, the complement, and, or, union, intersection, necessity and possibility operations were defined on interval-valued intuitionistic fuzzy soft sets, and the basic properties of interval-valued intuitionistic fuzzy soft sets were discussed. They [18] also presented an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and gave some illustrative examples, the weighted intuitionistic fuzzy soft sets were introduced and its application to decision making was investigated. Zhang [19] proposed a novel approach to intuitionistic fuzzy soft set based decision making problems using rough set theory. Wang and Qu [20] introduced the definitions of entropy, similarity measure and distance measure of vague soft sets, the relations between these measures are discussed in detail. However, there has been rather little work completed for decomposition theorems and representation theorems in the context of intuitionistic fuzzy soft sets. The purpose of this paper is to further extend the concept of vague soft set theory proposed by Xu et al. in [12]. In this paper, we will present the definitions of cut sets of vague soft sets and established the decomposition theorems and representation theorems.

The rest of this paper is organized as follows. In Section 2, we recall some basic definitions which will be used in the rest of the paper. In Section 3, we introduce the definitions of cut sets of vague soft sets, and their properties and relations are given. In Section 4, the decomposition theorems of vague soft sets are established. In Section 5, the representation theorems of vague soft sets are obtained. In the finial section, some concluding comments are presented.

2 Preliminaries

In this section, we will recall several definitions and results which are necessary for our paper. They are stated as follows:

Definition 2.1. [8] A vague set A in the universe U = {x₁, x₂, . . . , x_n} can be expressed by the following notion, A = {(x_i, [t_A (x_i) , 1 - f_A (x_i)]) |x_i ∈ U}, i.e A (x_i) = [t_A (x_i) , 1 - f_A (x_i)] and the condition 0 ≤ t_A (x_i) ≤1 - f_A (x_i) should hold for any x_i ∈ U, where t_A (x_i) is called the membership degree (true membership) of element x_i to the vague set A, while f_A (x_i) is the degree of nonmembership (false membership) of the element x_i to the vague set A.

Definition 2.2. [8] Let A, B be two vague sets in the universe U = {x₁, x₂, . . . , x_n}. If ∀x_i ∈ U, t_A (x_i) ≤ t_B (x_i) , 1 - f_A (x_i) ≤1 - f_B (x_i), then A is called a vague subset of B, denoted by A ⊆ B, where i = 1, 2, 3, . . . , n.

Definition 2.3. [10] Let U be an initial universe set, P (U) the power set of U, E a set of parameters and A ⊆ E. A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P (U).

Definition 2.4. [12] Let U be an initial universe set, V (U) the set of all vague sets on U, E a set of parameters and A ⊆ E. A pair (F, A) is called a vague soft set over U, where F is a mapping given by F : A → V (U).

Definition 2.5. [12] Let (F, A) and (G, B) be two vague soft sets over a universe U. If A ⊆ B and ∀e ∈ A, F (e) is a vague subset of G (e), then (F, A) is called a vague soft subset of (G, B). This relation is denoted by (F, A) ⊆ (G, B).

Definition 2.6. [12] Two vague soft sets (F, A) and (G, B) over a universe U are said to be vague soft equal if (F, A) is a vague soft subset of (G, B) and (G, B) is a vague soft subset of (F, A). This relation is denoted by (F, A) = (G, B).

Definition 2.7. [12] Let E = {e₁, e₂, . . . , e_n} be a parameter set. The not set of E denoted by ¬E is defined by ¬E = {¬ e₁, ¬ e₂, . . . , ¬ e_n}, where ¬e_i = not e_i.

Definition 2.8. [12] The complement of vague soft set (F, A) is denoted by (F, A) ^c and is defined by (F, A) ^c = (F^c, ¬ A), where F^c : ¬ A → V (U) is a mapping given by t_F^c(¬α) (x) = f_F(α) (x) , 1 - f_F^c(¬α) (x) =1 - t_F(α) (x) , ∀ ¬ α ∈ ¬ A, x ∈ U.

Clearly (F^c) ^c is the same as F and ((F, A) ^c) ^c = (F, A).

Definition 2.9. [12] The union of two vague soft sets (F, A) and (G, B) over a universe U is a vague soft set (H, C), where C = A ∪ B and ∀e ∈ C, $t_{H (e)} (x) = {\begin{matrix} t_{F (e)} (x) & e \in A - B, x \in U, \\ t_{G (e)} (x) & e \in B - A, x \in U, \\ t_{F (e)} (x) \lor t_{G (e)} (x) & e \in A \cap B, x \in U . \end{matrix}$ $\begin{matrix} 1 - f_{H (e)} (x) \\ = {\begin{matrix} 1 - f_{F (e)} (x) & e \in A - B, x \in U, \\ 1 - f_{G (e)} (x) & e \in B - A, x \in U, \\ (1 - f_{F (e)} (x)) & e \in A \cap B, x \in U . \\ \lor (1 - f_{G (e)} (x)) \end{matrix} \end{matrix}$

We denote it by (F, A) ∪ (G, B) = (H, C).

Definition 2.10. [12] The intersection of two vague soft sets (F, A) and (G, B) over a universe U is a vague soft set (H, C), where C = A ∪ B and ∀e ∈ C, $t_{H (e)} (x) = {\begin{matrix} t_{F (e)} (x) & e \in A - B, x \in U, \\ t_{G (e)} (x) & e \in B - A, x \in U, \\ t_{F (e)} (x) \land t_{G (e)} (x) & e \in A \cap B, x \in U . \end{matrix}$

$\begin{matrix} 1 - f_{H (e)} (x) \\ = {\begin{matrix} 1 - f_{F (e)} (x) & e \in A - B, x \in U, \\ 1 - f_{G (e)} (x) & e \in B - A, x \in U, \\ (1 - f_{F (e)} (x)) & e \in A \cap B, x \in U . \\ \land (1 - f_{G (e)} (x)) \end{matrix} \end{matrix}$

We denote it by (F, A) ∩ (G, B) = (H, C).

Definition 2.11. [11] Let U be an initial universe set, F (U) the set of all fuzzy sets on U, E a set of parameters and A ⊆ E. A pair (F, A) is called a fuzzy soft set over U, where F is a mapping given by F : A → F (U).

Definition 2.12. [11] A fuzzy soft set (F, A) over U is said to be a null fuzzy soft set denoted by $\tilde{\emptyset}$ , if ∀e ∈ A, F (e) (x) =0, x ∈ X.

Definition 2.13. [11] A fuzzy soft set (F, A) over U is said to be an absolute fuzzy soft set denoted by $\tilde{X}$ , if ∀e ∈ E, F (e) (x) =1, x ∈ X.

Definition 2.14. [4] Let X be a set. We call the mapping $A : X \to {0, \frac{1}{2}, 1}$ a triple valued fuzzy set. We denote the set of all triple valued fuzzy sets over X as 3^X.

3 The cut sets of vague soft sets

In this section, based on the definitions of cut sets of intuitionistic fuzzy sets and interval valued fuzzy sets [4], the concepts of cut sets of vague soft sets are introduced. Furthermore, some properties of these cut sets and their relations are given. Let X be an initial universe set and E be the non-empty set of parameters, we denote the set of all vague soft sets over X via parameters in E as VSS (X).

Definition 3.1. Let X be an initial universe set, 3^X the set of all triple valued fuzzy sets on X, E a set of parameters. A pair (F, E) is called a triple valued fuzzy soft set over X, where F is a mapping given by F : E → 3^X. We denote the set of all triple valued fuzzy soft sets over X via parameters in E as $FS (3^{X})$ .

Clearly, triple valued fuzzy soft set is a special kind of fuzzy soft set.

Definition 3.2. Let (F, E) ∈ VSS (X) and λ ∈ [0, 1]. The λ-upper cut set and λ-strong upper cut set of (F, E) denoted by (F, E) _λ and $(F, E)_{\underline{λ}}$ , are defined by (F, E) _λ = (F_λ, E) and $(F, E)_{\underline{λ}} = (F_{\underline{λ}}, E)$ , where F_λ : E → 3^X, $F_{\underline{λ}} : E \to 3^{X}$ are mappings given by ∀e ∈ E, ∀x ∈ X, $\begin{matrix} F_{λ} (e) (x) = {\begin{matrix} 1 & t_{F (e)} (x) \geq λ \\ \frac{1}{2} & t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x) \\ 0 & λ > 1 - f_{F (e)} (x) \end{matrix}, \\ F_{\underline{λ}} (e) (x) = {\begin{matrix} 1 & t_{F (e)} (x) > λ \\ \frac{1}{2} & t_{F (e)} (x) \leq λ < 1 - f_{F (e)} (x) \\ 0 & λ \geq 1 - f_{F (e)} (x) \end{matrix} . \end{matrix}$

Definition 3.3. Let (F, E) ∈ VSS (X) and λ ∈ [0, 1]. The λ-lower cut set and λ-strong lower cut set of (F, E) denoted by (F, E) ^λ and $(F, E)^{\underline{λ}}$ , are defined by (F, E) ^λ = (F^λ, E) and $(F, E)^{\underline{λ}} = (F^{\underline{λ}}, E)$ , where F^λ : E → 3^X, $F^{\underline{λ}} : E \to 3^{X}$ are mappings given by ∀e ∈ E, ∀x ∈ X, $\begin{matrix} F^{λ} (e) (x) = {\begin{matrix} 1 & 1 - f_{F (e)} (x) \leq λ \\ \frac{1}{2} & t_{F (e)} (x) \leq λ < 1 - f_{F (e)} (x) \\ 0 & t_{F (e)} (x) > λ \end{matrix}, \\ F^{\underline{λ}} (e) (x) = {\begin{matrix} 1 & 1 - f_{F (e)} (x) < λ \\ \frac{1}{2} & t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x) \\ 0 & t_{F (e)} (x) \geq λ \end{matrix} . \end{matrix}$

Definition 3.4. Let (F, E) ∈ VSS (X) and λ ∈ [0, 1]. The λ-upper quasi cut set and λ-strong upper quasi cut set of (F, E) denoted by (F, E) _[λ] and $(F, E)_{\underline{[λ]}}$ , are defined by (F, E) _[λ] = (F_[λ], E) and $(F, E)_{\underline{[λ]}} = (F_{\underline{[λ]}}, E)$ , where F_[λ] : E → 3^X, $F_{\underline{[λ]}} : E \to 3^{X}$ are mappings given by ∀e ∈ E, ∀x ∈ X, $\begin{matrix} F_{[λ]} (e) (x) = {\begin{matrix} 1 & λ + t_{F (e)} (x) \geq 1 \\ \frac{1}{2} & t_{F (e)} (x) < 1 - λ \leq 1 - f_{F (e)} (x) \\ 0 & λ < f_{F (e)} (x) \end{matrix}, \\ F_{\underline{[λ]}} (e) (x) = {\begin{matrix} 1 & λ + t_{F (e)} (x) > 1 \\ \frac{1}{2} & t_{F (e)} (x) \leq 1 - λ < 1 - f_{F (e)} (x) \\ 0 & λ \leq f_{F (e)} (x) \end{matrix} . \end{matrix}$

Definition 3.5. Let (F, E) ∈ VSS (X) and λ ∈ [0, 1]. The λ-lower quasi cut set and λ-strong lower quasi cut set of (F, E) denoted by (F, E) ^[λ] and $(F, E)^{\underline{[λ]}}$ , are defined by (F, E) ^[λ] = (F^[λ], E) and $(F, E)^{\underline{[λ]}} = (F^{\underline{[λ]}}, E)$ , where F^[λ] : E → 3^X, $F^{\underline{[λ]}} : E \to 3^{X}$ are mappings given by ∀e ∈ E, ∀x ∈ X, $\begin{matrix} F^{[λ]} (e) (x) = {\begin{matrix} 1 & λ \leq f_{F (e)} (x) \\ \frac{1}{2} & t_{F (e)} (x) \leq 1 - λ < 1 - f_{F (e)} (x) \\ 0 & λ + t_{F (e)} (x) > 1 \end{matrix}, \\ F^{\underline{[λ]}} (e) (x) = {\begin{matrix} 1 & λ < f_{F (e)} (x) \\ \frac{1}{2} & t_{F (e)} (x) < 1 - λ \leq 1 - f_{F (e)} (x) \\ 0 & λ + t_{F (e)} (x) \geq 1 \end{matrix} . \end{matrix}$

Clearly, the cut sets (F, E) _λ, $(F, E)_{\underline{λ}}$ , (F, E) ^λ, $(F, E)^{\underline{λ}}$ , (F, E) _[λ], $(F, E)_{\underline{[λ]}}$ , (F, E) ^[λ] and $(F, E)^{\underline{[λ]}}$ of vague soft set (F, E) are triple valued fuzzy soft sets. Next, the properties of these kinds of cut sets on vague soft sets are given.

Property 3.1.

$(F, E)_{\underline{λ}} \subseteq (F, E)_{λ}$ .

$(F, E)_{0} = \tilde{X}$ , $(F, E)_{\underline{1}} = \tilde{\emptyset}$ .

(F, E) ⊆ (G, E) ⇒ (F, E) _λ ⊆ (G, E) _λ, $(F, E)_{\underline{λ}} \subseteq (G, E)_{\underline{λ}}$ .

λ₁ < λ₂ ⇒ (F, E) _λ
₁ ⊇ (F, E) _λ
₂, $(F, E)_{\underline{λ_{1}}} \supseteq (F, E)_{\underline{λ_{2}}}$ , $(F, E)_{\underline{λ_{1}}} \supseteq (F, E)_{λ_{2}}$ .

((F, E) ∪ (G, E)) _λ = (F, E) _λ ∪ (G, E) _λ, $((F, E) \cup (G, E))_{\underline{λ}} = (F, E)_{\underline{λ}} \cup (G, E)_{\underline{λ}}$ , ((F, E) ∩ (G, E)) _λ = (F, E) _λ ∩ (G, E) _λ, $((F, E) \cap (G, E))_{\underline{λ}} = (F, E)_{\underline{λ}} \cap (G, E)_{\underline{λ}}$ .

Let λ_t ∈ [0, 1] (t ∈ T) , a = ⋀ _t∈Tλ_t, b = ⋁ _t∈Tλ_t. Then we have $⋃_{t \in T} (F, E)_{λ_{t}} \subseteq (F, E)_{a}, ⋃_{t \in T} (F, E)_{\underline{λ_{t}}} = (F, E)_{\underline{a}}, ⋂_{t \in T} (F, E)_{λ_{t}} = (F, E)_{b}, ⋂_{t \in T} (F, E)_{\underline{λ_{t}}} \supseteq (F, E)_{\underline{b}} .$

(⋃ _r∈Γ (F_r, E)) _λ ⊇ ⋃ _r∈Γ (F_r, E) _λ, $(⋃_{r \in Γ} (F_{r}, E))_{\underline{λ}} = ⋃_{r \in Γ} (F_{r}, E)_{\underline{λ}}$ , (⋂ _r∈Γ (F_r, E)) _λ = ⋂ _r∈Γ (F_r, E) _λ, $(⋂_{r \in Γ} (F_{r}, E))_{\underline{λ}} \subseteq ⋂_{r \in Γ} (F_{r}, E)_{\underline{λ}}$ .

$((F, E)^{c})_{λ} = ((F, E)_{\underline{1 - λ}})^{c}, ((F, E)^{c})_{\underline{λ}} = ((F, E)_{1 - λ})^{c}$ .

Proof. (1)–(2) are obvious.

(3) (F, E) ⊆ (G, E) ⇒ ∀ e ∈ E, F (e) ⊆ G (e) ⇒ ∀ e ∈ E, ∀ x ∈ X, t_F(e) (x) ≤ t_G(e) (x) , 1 - f_F(e) (x) ≤1 - f_G(e) (x) ⇒ F_λ (e) (x) ≤ G_λ (e) (x) ⇒ (F, E) _λ ⊆ (G, E) _λ. $(F, E)_{\underline{λ}} \subseteq (G, E)_{\underline{λ}}$ can be proved similarly.

(4) λ₁< λ₂ ⇒ For ∀e ∈ E, ∀ x ∈ X, when λ₁ < λ₂ ≤ t_F(e) (x), then (F, E) _{λ
₁} = (F, E) _{λ
₂}, when λ₁ ≤ t_F(e) (x) < λ₂ < 1, then (F, E) _{λ
₁} ⊇ (F, E) _{λ
₂}, when t_F(e) (x) < λ₁ < λ₂ ≤ 1 - f_F(e) (x), then (F, E) _{λ
₁} = (F, E) _{λ
₂}, when t_F(e) (x) < λ₁ ≤ 1 - f_F(e) (x) < λ₂ < 1, then (F, E) _{λ
₁} ⊇ (F, E) _{λ
₂}, when 1 - f_F(e) (x) < λ₁ < λ₂ ≤ 1, then (F, E) _{λ
₁} = (F, E) _{λ
₂}. Hence λ₁ < λ₂ ⇒ (F, E) _{λ
₁} ⊇ (F, E) _{λ
₂}. With the same reason, we have $λ_{1} < λ_{2} \Rightarrow (F, E)_{\underline{λ_{1}}} \supseteq (F, E)_{\underline{λ_{2}}}$ , $(F, E)_{\underline{λ_{1}}} \supseteq (F, E)_{λ_{2}}$ .

(5) Let (F, E) ∪ (G, E) = (H, E). ∀e ∈ E, ∀ x ∈ X, when H_λ (e) (x) =1 ⇔ t_H(e) (x) ≥ λ⇔t_F (e) (x) ⋁ t_G(e) (x) ≥ λ ⇔ t_F(e) (x) ≥ λ or t_G(e) (x) ≥ λ ⇔ F_λ (e) (x) =1 or G_λ (e) (x) =1 ⇔ (F_λ (e) ⋃G_λ (e)) (x) =1, when H_λ (e) (x) =0 ⇔ λ > 1 - f_H(e) (x) ⇔ λ > (1 - f_F(e) (x)) ⋁ (1 - f_G(e) (x)) ⇔1 - f_F(e) (x) < λ and 1 - f_G(e) (x) < λ ⇔ F_λ (e) (x) =0 and G_λ (e) (x) =0 ⇔ (F_λ (e) ⋃ G_λ (e)) (x) =0. Since $(H, E)_{λ} = ((F, E) \cup (G, E))_{λ}, (F, E)_{λ}, (G, E)_{λ} \in FS (3^{X})$ , we have ((F, E) ∪ (G, E)) _λ = (F, E) _λ ∪ (G, E) _λ. Others can be proved similarly.

(6) For ∀e ∈ E, ∀ x ∈ X, ⋁_t∈TF_{λ
_t} (e) (x) =1 ⇒ ∃t ∈ T, F_{λ
_t} (e) (x) =1 ⇒ ∃t ∈ T, t_F(e) (x) ≥ λ_t ≥ a ⇒ F_a (e) (x) =1.

For ∀e ∈ E, ∀ x ∈ X, F_a (e) (x) =0 ⇒ a = ⋀ _t∈Tλ_t > 1 - f_F(e) (x) ⇒ ∀t ∈ T, λ_t > 1 - f_F(e) (x) ⇒ ∀t ∈ T, F_{λ
_t} (e) (x) =0 ⇒ ⋁_t∈TF_{λ
_t} (e) (x) =0.

Since ⋃_t∈T (F, E) _{λ
_t}, $(F, E)_{a} \in FS (3^{X})$ , we have ⋃_t∈T (F, E) _{λ
_t} ⊆ (F, E) _a.

For ∀e ∈ E, ∀ x ∈ X, $⋁_{t \in T} F_{\underline{λ_{t}}} (e) (x) = 1$ ⇔ $\exists t \in T, F_{\underline{λ_{t}}} (e) (x) = 1$ ⇔ ∃t ∈ T, t_F(e) (x) > λ_t ≥ a ⇔ F_a (e) (x) =1.

For ∀e ∈ E, ∀ x ∈ X, $⋁_{t \in T} F_{\underline{λ_{t}}} (e) (x) = 0$ ⇔ ∀t ∈ T, λ_t > 1 - f_F(e) (x) ⇔ a ≥ 1 - f_F(e) (x) ⇔ $F_{\underline{a}} (e) (x) = 0$ .

Since $⋃_{t \in T} (F, E)_{\underline{λ_{t}}}$ , $(F, E)_{\underline{a}} \in FS (3^{X})$ , we have $⋃_{t \in T} (F, E)_{\underline{λ_{t}}} = (F, E)_{\underline{a}}$ .

(7) Let ⋃_r∈Γ (F_r, E) = (H, E). H_λ (e) (x) =1 ⇒ ⋁ _r∈Γ (F_r) _λ (e) (x) =1 ⇒ ∃ r ∈ Γ, (F_r) _λ (e) (x) =1 ⇒ ∃ r ∈ Γ, t_{(F_r)(e)} (x) ≥ λ ⇒ ⋁ _r∈Γt_{(F_r)(e)} (x) ≥ λ ⇒ ⋃ _r∈Γ (F_r) _λ (e) (x) =1.

H_λ (e) (x) =0 ⇒ ⋁ _r∈Γ (F_r) _λ (e) (x) =0 ⇒ ∀ r ∈ Γ, (F_r) _λ (e) (x) =0 ⇒ ∀ r ∈ Γ, λ > 1 - f_{(F_r)(e)} (x) ⇒ ⋁ _r∈Γ (1 - f_{(F_r)(e)} (x)) < λ ⇒ ⋃ _r∈Γ (F_r) _λ (e) (x) =0.

Since (⋃ _r∈Γ (F_r, E)) _λ, $⋃_{r \in Γ} (F_{r}, E)_{λ} \in FS (3^{X})$ , we have (⋃ _r∈Γ (F_r, E)) _λ ⊇ ⋃ _r∈Γ (F_r, E) _λ.

$H_{\underline{λ}} (e) (x) = 1 \Leftrightarrow ⋁_{r \in Γ} (F_{r})_{\underline{λ}} (e) (x) = 1 \Leftrightarrow \exists r \in Γ, (F_{r})_{\underline{λ}} (e) (x) = 1 \Leftrightarrow \exists r \in Γ, t_{(F_{r}) (e)} (x) > λ \Leftrightarrow ⋁_{r \in Γ} t_{(F_{r}) (e)} (x) > λ \Leftrightarrow ⋃_{r \in Γ} (F_{r})_{\underline{λ}} (e) (x) = 1$ .

$H_{\underline{λ}} (e) (x) = 0 \Leftrightarrow ⋁_{r \in Γ} (F_{r})_{\underline{λ}} (e) (x) = 0 \Leftrightarrow \forall r \in Γ, (F_{r})_{\underline{λ}} (e) (x) = 0 \Leftrightarrow \forall r \in Γ, λ \geq 1 - f_{(F_{r}) (e)} (x) \Leftrightarrow ⋁_{r \in Γ} (1 - f_{(F_{r}) (e)} (x)) \leq λ \Leftrightarrow ⋃_{r \in Γ} (F_{r})_{\underline{λ}} (e) (x) = 0$ .

Since $(⋃_{r \in Γ} (F_{r}, E))_{\underline{λ}}$ , $⋃_{r \in Γ} (F_{r}, E)_{\underline{λ}} \in FS (3^{X})$ , we have $(⋃_{r \in Γ} (F_{r}, E))_{\underline{λ}} = ⋃_{r \in Γ} (F_{r}, E)_{\underline{λ}}$ .

(8) (F, E) ^c = (F^c, ¬ E), ${(F^{c})}_{λ} (\neg e) (x) = 1 \Leftrightarrow_{F^{c} (\neg e)} (x) = f_{F (e)} (x) \geq λ \Leftrightarrow 1 - λ \geq 1 - f_{F (e)} (x) \Leftrightarrow F_{\underline{1 - λ}} (e) (x) = 0 \Leftrightarrow (F_{\underline{1 - λ}} (e))^{c} (x) = 1$ .

$(F^{c})_{λ} (\neg e) (x) = 0 \Leftrightarrow λ > 1 - f_{F^{c} (\neg e)} (x) = 1 - t_{F (e)} (x) \Leftrightarrow t_{F (e)} (x) > 1 - λ \Leftrightarrow F_{\underline{1 - λ}} (e) (x) = 0 \Leftrightarrow (F_{\underline{1 - λ}} (e))^{c} (x) = 1$ .

Since ((F, E) ^c) _λ, $((F, E)_{\underline{1 - λ}})^{c} \in FS (3^{X})$ , we have $((F, E)^{c})_{λ} = ((F, E)_{\underline{1 - λ}})^{c}$ . □

Property 3.2.

$(F, E)^{\underline{λ}} \subseteq (F, E)^{λ}$ .

$(F, E)^{\underline{0}} = \tilde{\emptyset}$ , $(F, E)^{1} = \tilde{X}$ .

(F, E) ⊆ (G, E) ⇒ (F, E) ^λ ⊇ (G, E) ^λ, $(F, E)^{\underline{λ}} \supseteq (G, E)^{\underline{λ}}$ .

λ₁ < λ₂ ⇒ (F, E) ^λ
₁ ⊆ (F, E) ^λ
₂, $(F, E)^{\underline{λ_{1}}} \subseteq (F, E)^{\underline{λ_{2}}}$ , $(F, E)^{λ_{1}} \subseteq (F, E)^{\underline{λ_{2}}}$ .

((F, E) ∪ (G, E)) ^λ = (F, E) ^λ ∪ (G, E) ^λ, $((F, E) \cup (G, E))^{\underline{λ}} = (F, E)^{\underline{λ}} \cup (G, E)^{\underline{λ}}$ , ((F, E) ∩ (G, E)) ^λ = (F, E) ^λ ∩ (G, E) ^λ, $((F, E) \cap (G, E))^{\underline{λ}} = (F, E)^{\underline{λ}} \cap (G, E)^{\underline{λ}}$ .

Let λ_t ∈ [0, 1] (t ∈ T), a = ⋀ _t∈Tλ_t, b = ⋁ _t∈Tλ_t. Then we have $⋃_{t \in T} (F, E)^{λ_{t}} \subseteq (F, E)^{b}, ⋃_{t \in T} (F, E)^{\underline{λ_{t}}} = (F, E)^{\underline{b}}, ⋂_{t \in T} (F, E)^{λ_{t}} = (F, E)^{a}, ⋂_{t \in T} (F, E)^{\underline{λ_{t}}} \supseteq (F, E)^{\underline{a}} .$

(⋃ _t∈T (F_t, E)) ^λ = ⋂ _t∈T (F_t, E) ^λ, $(⋃_{t \in T} (F_{t}, E))^{\underline{λ}} \subseteq ⋂_{t \in T} (F_{t}, E)^{\underline{λ}}$ , (⋂ _t∈T (F_t, E)) ^λ ⊇ ⋃ _t∈T (F_t, E) ^λ, $(⋂_{t \in T} (F_{t}, E))^{\underline{λ}} = ⋃_{t \in T} (F_{t}, E)^{\underline{λ}}$ .

$((F, E)^{c})^{λ} = ((F, E)^{\underline{1 - λ}})^{c}$ , $((F, E)^{c})^{\underline{λ}} = ((F, E)^{1 - λ})^{c}$ .

Property 3.3.

$(F, E)_{\underline{[λ]}} \subseteq (F, E)_{[λ]}$ .

$(F, E)_{[1]} = \tilde{X}$ , $(F, E)_{\underline{[0]}} = \tilde{\emptyset}$ .

(F, E) ⊆ (G, E) ⇒ (F, E) _[λ] ⊆ (G, E) _[λ], $(F, E)_{\underline{[λ]}} \subseteq (G, E)_{\underline{[λ]}}$ .

λ₁ < λ₂ ⇒ (F, E) _[λ₁] ⊆ (F, E) _[λ₂], $(F, E)_{\underline{[λ_{1}]}} \subseteq (F, E)_{\underline{[λ_{2}]}}$ , $(F, E)_{[λ_{1}]} \subseteq (F, E)_{\underline{[λ_{2}]}}$ .

((F, E) ∪ (G, E)) _[λ] = (F, E) _[λ] ∪ (G, E) _[λ], $((F, E) \cup (G, E))_{\underline{[λ]}} = (F, E)_{\underline{[λ]}} \cup (G, E)_{\underline{[λ]}}$ , ((F, E) ∩ (G, E)) _[λ] = (F, E) _[λ] ∩ (G, E) _[λ], $((F, E) \cap (G, E))_{\underline{[λ]}} = (F, E)_{\underline{[λ]}} \cap (G, E)_{\underline{[λ]}}$ .

Let λ_t ∈ [0, 1] (t ∈ T), a = ⋀ _t∈Tλ_t, b = ⋁ _t∈Tλ_t. Then we have $⋃_{t \in T} (F, E)_{[λ_{t}]} \subseteq (F, E)_{[b]}, ⋃_{t \in T} (F, E)_{\underline{[λ_{t}]}} = (F, E)_{\underline{[b]}}, ⋂_{t \in T} (F, E)_{[λ_{t}]} = (F, E)_{[a]}, ⋂_{t \in T} (F, E)_{\underline{[λ_{t}]}} \supseteq (F, E)_{\underline{[a]}} .$

(⋃ _t∈T (F_t, E)) _[λ] ⊇ ⋃ _t∈T (F_t, E) _[λ], $(⋃_{t \in T} (F_{t}, E))_{\underline{[λ]}} = ⋃_{t \in T} (F_{t}, E)_{\underline{[λ]}}$ , (⋂ _t∈T (F_t, E)) _[λ] = ⋂ _t∈T (F_t, E) _[λ], $(⋂_{t \in T} (F_{t}, E))_{\underline{[λ]}} \subseteq ⋂_{t \in T} (F_{t}, E)_{\underline{[λ]}}$ .

$((F, E)^{c})_{[λ]} = ((F, E)_{\underline{[1 - λ]}})^{c}$ , $((F, E)^{c})_{\underline{[λ]}} = ((F, E)_{[1 - λ]})^{c}$ .

Property 3.4.

$(F, E)^{\underline{[λ]}} \subseteq (F, E)^{[λ]}$ .

$(F, E)^{\underline{[1]}} = \tilde{\emptyset}$ , $(F, E)^{[0]} = \tilde{X}$ .

(F, E) ⊆ (G, E) ⇒ (F, E) ^[λ] ⊇ (G, E) ^[λ], $(F, E)^{\underline{[λ]}} \supseteq (G, E)^{\underline{[λ]}}$ .

λ₁ < λ₂ ⇒ (F, E) ^[λ₁] ⊇ (F, E) ^[λ₂], $(F, E)^{\underline{[λ_{1}]}} \supseteq (F, E)^{\underline{[λ_{2}]}}$ , $(F, E)^{\underline{[λ_{1}]}} \supseteq (F, E)^{[λ_{2}]}$ .

((F, E) ∪ (G, E)) ^[λ] = (F, E) ^[λ] ∪ (G, E) ^[λ], $((F, E) \cup (G, E))^{\underline{[λ]}} = (F, E)^{\underline{[λ]}} \cup (G, E)^{\underline{[λ]}}$ , ((F, E) ∩ (G, E)) ^[λ] = (F, E) ^[λ] ∩ (G, E) ^[λ], $((F, E) \cap (G, E))^{\underline{[λ]}} = (F, E)^{\underline{[λ]}} \cap (G, E)^{\underline{[λ]}}$ .

Let λ_t ∈ [0, 1] (t ∈ T), a = ⋀ _t∈Tλ_t, b = ⋁ _t∈Tλ_t. Then we have $⋃_{t \in T} (F, E)^{[λ_{t}]} \subseteq (F, E)^{[a]}, ⋃_{t \in T} (F, E)^{\underline{[λ_{t}]}} = (F, E)^{\underline{[a]}}, ⋂_{t \in T} (F, E)^{[λ_{t}]} = (F, E)^{[b]}, ⋂_{t \in T} (F, E)^{\underline{[λ_{t}]}} \supseteq (F, E)^{\underline{[b]}} .$

(⋃ _t∈T (F_t, E)) ^[λ] = ⋂ _t∈T (F_t, E) ^[λ], $(⋃_{t \in T} (F_{t}, E))^{\underline{[λ]}} \subseteq ⋂_{t \in T} (F_{t}, E)^{\underline{[λ]}}$ , (⋂ _t∈T (F_t, E)) ^[λ] ⊇ ⋃ _t∈T (F_t, E) ^[λ], $(⋂_{t \in T} (F_{t}, E))^{\underline{[λ]}} = ⋃_{t \in T} (F_{t}, E)^{\underline{[λ]}}$ .

$((F, E)^{c})^{[λ]} = ((F, E)^{\underline{[1 - λ]}})^{c}$ , $((F, E)^{c})^{\underline{[λ]}} = ((F, E)^{[1 - λ]})^{c}$ .

The proofs of Properties 3.2–3.4 are similar to that of Property 3.1.

Remark 3.1.

From Properties 3.1–3.4, we can see that each kind of cut set on vague soft sets has completely similar properties.

By the results in [3 –6], we can know that the cut set of vague soft sets has the same properties as that of fuzzy sets, intuitionistic fuzzy sets, interval valued fuzzy sets or three dimensional fuzzy sets.

We have the relations among these cut sets of vague soft sets as follows:

Theorem 3.1.

(F, E) _λ = ((F, E) ^c) ^1-λ = (F, E) _[1-λ] = ((F, E) ^c) ^[λ].

$(F, E)_{\underline{λ}} = ((F, E)^{c})^{\underline{1 - λ}} = (F, E)_{\underline{[1 - λ]}} = ((F, E)^{c})^{\underline{[λ]}}$ .

(F, E) ^λ = ((F, E) ^c) _[λ] = (F, E) ^[1-λ] = ((F, E) ^c) _1-λ.

$(F, E)^{\underline{λ}} = ((F, E)^{c})_{\underline{[λ]}} = (F, E)^{\underline{[1 - λ]}} = ((F, E)^{c})_{\underline{1 - λ}}$ .

(F, E) _[λ] = (F, E) _1-λ, $(F, E)_{\underline{[λ]}} = (F, E)_{\underline{1 - λ}}$ .

$(F, E)^{λ} = ((F, E)_{\underline{λ}})^{c}$ , $(F, E)^{\underline{λ}} = ((F, E)_{λ})^{c}$

$(F, E)^{[λ]} = ((F, E)_{\underline{1 - λ}})^{c}, (F, E)^{\underline{[λ]}} = ((F, E)_{1 - λ})^{c}$ .

$(F, E)_{λ} = ((F, E)^{\underline{[1 - λ]}})^{c}, (F, E)_{\underline{λ}} = ((F, E)^{[1 - λ]})^{c}$ .

$(F, E)^{λ} = ((F, E)_{\underline{[1 - λ]}})^{c}, (F, E)^{\underline{λ}} = ((F, E)_{[1 - λ]})^{c}$ .

$(F, E)_{λ} = ((F, E)^{\underline{λ}})^{c}, (F, E)^{λ} = ((F, E)_{\underline{λ}})^{c}$ .

$(F, E)_{[λ]} = ((F, E)^{\underline{[λ]}})^{c}, (F, E)^{[λ]} = ((F, E)_{\underline{[λ]}})^{c}$ .

Proof. We only need to prove the cases (1) and (5), others can be proved similarly.

(1) For ∀e ∈ E, ∀ x ∈ X, $\begin{matrix} F_{λ} (e) (x) \\ = {\begin{matrix} 1 & t_{F (e)} (x) \geq λ \\ \frac{1}{2} & t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x) \\ 0 & λ > 1 - f_{F (e)} (x) \end{matrix} \\ = {\begin{matrix} 1 & 1 - λ \geq 1 - t_{F (e)} (x) \\ \frac{1}{2} & f_{F (e)} (x) \leq 1 - λ < 1 - t_{F (e)} (x) \\ 0 & 1 - λ < f_{F (e)} (x) \end{matrix} \\ = {\begin{matrix} 1 & 1 - λ \geq 1 - f_{F^{c} (\neg e)} (x) \\ \frac{1}{2} & t_{F^{c} (\neg e)} (x) \leq 1 - λ < 1 - f_{F^{c} (\neg e)} (x) \\ 0 & 1 - λ < t_{F^{c} (\neg e)} (x) \end{matrix} \\ = (F^{c})^{1 - λ} (\neg e) (x) \\ = {\begin{matrix} 1 & 1 - λ + t_{F (e)} (x) \geq 1 \\ \frac{1}{2} & t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x) \\ 0 & 1 - λ < f_{F (e)} (x) \end{matrix} \\ = F_{[1 - λ]} (e) (x) \\ = {\begin{matrix} 1 & λ \leq t_{F (e)} (x) \\ \frac{1}{2} & f_{F (e)} (x) \leq 1 - λ < 1 - t_{F (e)} (x) \\ 0 & 1 - λ < f_{F (e)} (x) \end{matrix} \\ = {\begin{matrix} 1 & λ \leq f_{F^{c} (\neg e)} (x) \\ \frac{1}{2} & t_{F^{c} (\neg e)} (x) \leq 1 - λ < 1 - f_{F^{c} (\neg e)} (x) \\ 0 & λ + t_{F^{c} (\neg e)} (x) > 1 \end{matrix} \\ = (F^{c})^{[λ]} (\neg e) (x) . \end{matrix}$

Hence (F, E) _λ = ((F, E) ^c) ^1-λ = (F, E) _[1-λ] = ((F, E) ^c) ^[λ].

(5) For ∀e ∈ E, ∀ x ∈ X, $\begin{matrix} F_{[λ]} (e) (x) \\ = {\begin{matrix} 1 & λ + t_{F (e)} (x) \geq 1 \\ \frac{1}{2} & t_{F (e)} (x) < 1 - λ \leq 1 - f_{F (e)} (x) \\ 0 & λ < f_{F (e)} (x) \end{matrix} \\ = {\begin{matrix} 1 & t_{F (e)} (x) \geq 1 - λ \\ \frac{1}{2} & t_{F (e)} (x) < 1 - λ \leq 1 - f_{F (e)} (x) \\ 0 & 1 - λ > 1 - f_{F (e)} (x) \end{matrix} \\ = F_{1 - λ} (e) (x), Hence (F, E)_{[λ]} \\ = (F, E)_{1 - λ} . For \forall e \in E, \forall x \in X, \\ F_{\underline{[λ]}} (e) (x) \\ = {\begin{matrix} 1 & λ + t_{F (e)} (x) > 1 \\ \frac{1}{2} & t_{F (e)} (x) \leq 1 - λ < 1 - f_{F (e)} (x) \\ 0 & λ \leq f_{F (e)} (x) \end{matrix} \\ = {\begin{matrix} 1 & t_{F (e)} (x) > 1 - λ \\ \frac{1}{2} & t_{F (e)} (x) \leq 1 - λ < 1 - f_{F (e)} (x) \\ 0 & 1 - λ \geq 1 - f_{F (e)} (x) \end{matrix} \\ = F_{\underline{1 - λ}} (e) (x), Hence (F, E)_{\underline{[λ]}} \\ = (F, E)_{\underline{1 - λ}} . □ \end{matrix}$

4 The decomposition theorems of vague soft sets

In this section, we will obtain the decomposition theorems of vague soft sets.

Definition 4.1. Let $(G, E) \in FS (3^{X}), λ \in [0, 1]$ . We define λ (G, E) = (λG, E) , λ · (G, E) = (λ·G, E) , λ ∗ (G, E) = (λ ∗ G, E), λ ⋄ (G, E) = (λ⋄G, E) , λ ★ (G, E) = (λ ★ G, E) , λ ∘ (G, E) = (λ○G, E), λ ⊗ (G, E) = (λ ⊗ G, E), λ ⊙ (G, E) = (λ ⊙ G, E) as vague soft sets over X, where ∀e ∈ E, ∀x ∈ X, $\begin{matrix} λ G (e) (x) & = & {\begin{matrix} [0, 0] & G (e) (x) = 0 \\ [0, λ] & G (e) (x) = \frac{1}{2} \\ [λ, λ] & G (e) (x) = 1 \end{matrix}, \\ λ \cdot G (e) (x) & = & {\begin{matrix} [λ, λ] & G (e) (x) = 0 \\ [λ, 1] & G (e) (x) = \frac{1}{2} \\ [1, 1] & G (e) (x) = 1 \end{matrix}, \\ λ * G (e) (x) & = & {\begin{matrix} [1 - λ, 1 - λ] & G (e) (x) = 0 \\ [0, 1 - λ] & G (e) (x) = \frac{1}{2} \\ [0, 0] & G (e) (x) = 1 \end{matrix}, \\ λ ⋄ G (e) (x) & = & {\begin{matrix} [1, 1] & G (e) (x) = 0 \\ [1 - λ, 1] & G (e) (x) = \frac{1}{2} \\ [1 - λ, 1 - λ] & G (e) (x) = 1 \end{matrix}, \\ λ ★ G (e) (x) & = & {\begin{matrix} [0, 0] & G (e) (x) = 0 \\ [0, 1 - λ] & G (e) (x) = \frac{1}{2} \\ [1 - λ, 1 - λ] & G (e) (x) = 1 \end{matrix}, \\ λ \circ G (e) (x) & = & {\begin{matrix} [1 - λ, 1 - λ] & G (e) (x) = 0 \\ [1 - λ, 1] & G (e) (x) = \frac{1}{2} \\ [1, 1] & G (e) (x) = 1 \end{matrix}, \\ λ \otimes G (e) (x) & = & {\begin{matrix} [λ, λ] & G (e) (x) = 0 \\ [0, λ] & G (e) (x) = \frac{1}{2} \\ [0, 0] & G (e) (x) = 1 \end{matrix}, \\ λ ⊙ G (e) (x) & = & {\begin{matrix} [1, 1] & G (e) (x) = 0 \\ [λ, 1] & G (e) (x) = \frac{1}{2} \\ [λ, λ] & G (e) (x) = 1 \end{matrix} . \end{matrix}$

By the Definition 4.1, we have the property as follows:

Property 4.1. Let $(G, E), (F, E) \in FS (3^{X})$ , λ ∈ [0, 1]. Then

λ₁ < λ₂ ⇒ λ₁ (G, E) ⊆ λ₂ (G, E), λ₁ · (G, E) ⊆ λ₂ · (G, E) , λ₁ ⊗ (G, E) ⊆ λ₂ ⊗ (G, E) , λ₁ ⊙ (G, E) ⊆ λ₂ ⊙ (G, E) , λ₁ ∗ (G, E) ⊇ λ₂ ∗ (G, E) , λ₁ ⋄ (G, E) ⊇ λ₂ ⋄ (G, E) , λ₁ ★ (G, E) ⊇ λ₂ ★ (G, E) , λ₁ ∘ (G, E) ⊇ λ₂ ∘ (G, E).

(F, E) ⊆ (G, E) ⇒ λ (F, E) ⊆ λ (G, E) , λ · (F, E) ⊆ λ · (G, E) , λ ★ (F, E) ⊆ λ ★ (G, E) , λ ∘ (F, E) ⊆ λ ∘ (G, E) , λ ∗ (F, E) ⊇ λ ∗ (G, E) , λ ⋄ (F, E) ⊇ λ ⋄ (G, E) , λ ⊗ (F, E) ⊇ λ ⊗ (G, E) , λ ⊙ (F, E) ⊇ λ ⊙ (G, E).

Then we have the following decomposition theorems of vague soft sets.

Theorem 4.1. Let (F, E) be a vague soft set over X.

(F, E) = ⋃ _λ∈[0,1]λ (F, E) _λ = ⋂ _λ∈[0,1]λ · (F, E) _λ.

$(F, E) = ⋃_{λ \in [0, 1]} λ (F, E)_{\underline{λ}} = ⋂_{λ \in [0, 1]} λ \cdot (F, E)_{\underline{λ}}$ .

If the mapping $H : [0, 1] \to FS (3^{X})$ satisfies $(F, E)_{\underline{λ}} \subseteq H (λ) \subseteq (F, E)_{λ}$ , then

(F, E) = ⋃ _λ∈[0,1]λH (λ) = ⋂ _λ∈[0,1]λ · H (λ).

λ₁ < λ₂ ⇒ H (λ₁) ⊇ H (λ₂).

$(F, E)_{λ} = ⋂_{α < λ} H (α), (F, E)_{\underline{λ}} = ⋃_{α > λ} H (α) .$

Proof. (1) $\forall e \in E, \forall x \in X, ⋁_{λ \in [0, 1]} λ F_{λ} (e) (x) = (⋁ {[λ, λ] | λ \in [0, 1] and F_{λ} (e) (x) = 1}) ⋁ (⋁ {[0, λ] | λ \in [0, 1] and F_{λ} (e) (x) = \frac{1}{2}}) = (⋁ {[λ, λ] | t_{F (e)} (x) \geq λ}) ⋁ (⋁ {[0, λ] | t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x)}) = [⋁_{λ \leq t_{F (e)} (x)} λ, ⋁_{λ \leq t_{F (e)} (x)} λ] ⋁ [0, ⋁_{t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x)} λ] = [t_{F (e)} (x), t_{F (e)} (x)] ⋁ [0, 1 - f_{F (e)} (x)] = [t_{F (e)} (x), 1 - f_{F (e)} (x)] = F (e) (x)$ . Hence (F, E) = ⋃ _λ∈[0,1]λ (F, E) _λ.

∀e ∈ E, $\forall x \in X, ⋀_{λ \in [0, 1]} λ \cdot F_{λ} (e) (x) = (⋀ {[λ, 1] | λ \in [0, 1] and F_{λ} (e) (x) = \frac{1}{2}}) ⋀ (⋀ {[λ, λ] | λ \in [0, 1] and F_{λ} (e) (x) = 0}) = (⋀ {[λ, 1] | t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x)}) ⋀ (⋀ {[λ, λ] | λ > 1 - f_{F (e)} (x)}) = [⋀_{t_{F (e)} (x) < λ \leq 1 - f_{F (e)} (x)} λ, 1] ⋀ [⋀_{λ > 1 - f_{F (e)} (x)} λ, ⋀_{λ > 1 - f_{F (e)} (x)} λ] = [t_{F (e)} (x), 1] ⋀ [1 - f_{F (e)} (x), 1 - f_{F (e)} (x)] = [t_{F (e)} (x), 1 - f_{F (e)} (x)] = F (e) (x)$ .

Hence (F, E) = ⋂ _λ∈[0,1]λ · (F, E) _λ.

(2) The proof is similar to (1).

(3) (a) Since $(F, E)_{\underline{λ}} \subseteq H (λ) \subseteq (F, E)_{λ}$ , so $λ (F, E)_{\underline{λ}} \subseteq λ H (λ) \subseteq λ (F, E)_{λ}, λ \cdot (F, E)_{\underline{λ}} \subseteq λ \cdot H (λ) \subseteq λ \cdot (F, E)_{λ}$ . Then $(F, E) = ⋃_{λ \in [0, 1]} λ (F, E)_{\underline{λ}} \subseteq ⋃_{λ \in [0, 1]} λ H (λ) \subseteq ⋃_{λ \in [0, 1]} λ (F, E)_{λ} = (F, E), (F, E) = ⋂_{λ \in [0, 1]} λ \cdot (F, E)_{\underline{λ}} \subseteq ⋂_{λ \in [0, 1]} λ \cdot H (λ) \subseteq ⋂_{λ \in [0, 1]} λ \cdot (F, E)_{λ} = (F, E) .$ Hence (F, E) = ⋃ _λ∈[0,1]λH (λ) = ⋂ _λ∈[0,1]λ · H (λ).

(b) $λ_{1} < λ_{2} \Rightarrow H (λ_{1}) \supseteq (F, E)_{\underline{λ_{1}}} \supseteq (F, E)_{λ_{2}} \supseteq H (λ_{2})$ , i.e. H (λ₁) ⊇ H (λ₂).

(c) If ∀α < λ, then $H (α) \supseteq (F, E)_{\underline{α}} \supseteq (F, E)_{λ}$ . Hence ⋂_α<λH (α) ⊇ (F, E) _λ. Since H (α) ⊆ (F, E) _α, so ⋂_α<λH (α) ⊆ ⋂ _α<λ (F, E) _α = (F, E) _λ. Hence (F, E) _λ = ⋂ _α<λH (α).

If ∀α > λ, then $(F, E)_{\underline{λ}} \supseteq (F, E)_{α} \supseteq H (λ)$ . Hence $(F, E)_{\underline{λ}} \supseteq ⋃_{α > λ} H (α)$ . Since $H (α) \supseteq (F, E)_{\underline{α}}$ , so $⋃_{α > λ} H (α) \supseteq ⋃_{α > λ} (F, E)_{\underline{α}} = (F, E)_{\underline{λ}}$ . Hence $(F, E)_{\underline{λ}} = ⋃_{α > λ} H (α)$ . □

Theorem 4.2. Let (F, E) be a vague soft set over X.

(F, E) = ⋃ _λ∈[0,1]λ ∗ (F, E) ^[λ] = ⋂ _λ∈[0,1]λ ⋄ (F, E) ^[λ].

$(F, E) = ⋃_{λ \in [0, 1]} λ * (F, E)^{\underline{[λ]}} = ⋂_{λ \in [0, 1]} λ ⋄ (F, E)^{\underline{[λ]}}$ .

If the mapping $H : [0, 1] \to FS (3^{X})$ satisfies $(F, E)^{\underline{[λ]}} \subseteq H (λ) \subseteq (F, E)^{[λ]}$ , then

(F, E) = ⋃ _λ∈[0,1]λ ∗ H (λ) = ⋂ _λ∈[0,1]λ ⋄ H (λ).

λ₁ < λ₂ ⇒ H (λ₁) ⊇ H (λ₂).

(F, E) ^[λ] = ⋂ _α<λH (α), $(F, E)^{\underline{[λ]}} = ⋃_{α > λ} H (α) .$

Theorem 4.3. Let (F, E) be a vague soft set over X.

(F, E) = ⋃ _λ∈[0,1]λ ★ (F, E) _[λ] = ⋂ _λ∈[0,1]λ ∘ (F, E) _[λ].

$(F, E) = ⋃_{λ \in [0, 1]} λ ★ (F, E)_{[\underline{λ}]} = ⋂_{λ \in [0, 1]} λ \circ (F, E)_{[\underline{λ}]}$ .

If the mapping $H : [0, 1] \to FS (3^{X})$ satisfies $(F, E)_{[\underline{λ}]} \subseteq H (λ) \subseteq (F, E)_{[λ]}$ , then

(F, E) = ⋃ _λ∈[0,1]λ ★ H (λ) = ⋂ _λ∈[0,1]λ ∘ H (λ).

λ₁ < λ₂ ⇒ H (λ₁) ⊆ H (λ₂).

(F, E) _[λ] = ⋂ _α>λH (α), $(F, E)_{[\underline{λ}]} = ⋃_{α < λ} H (α) .$

Theorem 4.4. Let (F, E) be a vague soft set over X.

(F, E) = ⋃ _λ∈[0,1]λ ⊗ (F, E) ^λ = ⋂ _λ∈[0,1]λ ⊙ (F, E) ^λ.

$(F, E) = ⋃_{λ \in [0, 1]} λ \otimes (F, E)^{\underline{λ}} = ⋂_{λ \in [0, 1]} λ ⊙ (F, E)^{\underline{λ}}$ .

If the mapping $H : [0, 1] \to FS (3^{X})$ satisfies $(F, E)^{\underline{λ}} \subseteq H (λ) \subseteq (F, E)^{λ}$ , then

(F, E) = ⋃ _λ∈[0,1]λ ⊗ H (λ) = ⋂ _λ∈[0,1]λ ⊙ H (λ).

λ₁ < λ₂ ⇒ H (λ₁) ⊆ H (λ₂).

(F, E) ^λ = ⋂ _α>λH (α), $(F, E)^{\underline{λ}} = ⋃_{α < λ} H (α) .$

The proofs of Theorems 4.2–4.4 are similar to that of Theorem 4.1.

5 The representation theorems of vague soft sets

Definition 5.1. Let $H : [0, 1] \to FS (3^{X})$ be a mapping. If λ₁ < λ₂ ⇒ H (λ₁) ⊇ H (λ₂), then H is called a inverse order nested set on X. We use $U (X)$ to denote the set of all inverse order nested sets of X.In $U (X)$ , we define

H₁ ⊆ H₂ ⇔ H₁ (α) ⊆ H₂ (α) , α ∈ [0, 1].

⋃_t∈TH_t : (⋃ _t∈TH_t) (α) = ⋃ _t∈TH_t (α) , α ∈ [0, 1].

⋂_t∈TH_t : (⋂ _t∈TH_t) (α) = ⋂ _t∈TH_t (α) , α ∈ [0, 1].

H^c : H^c (α) = (H (1 - α)) ^c, α ∈ [0, 1].

Theorem 5.1. Let $T_{i} : U (X) \to VSS (X)$ be a mapping (i = 1, 2), where T₁ (H) = ⋃ _λ∈[0,1]λH (λ), T₂ (H) = ⋂ _λ∈[0,1]λ · H (λ). Then

T₁ (H) = T₂ (H).

$T_{1} (H)_{λ} = ⋂_{α < λ} H (α), T_{1} (H)_{\underline{λ}} = ⋃_{α > λ} H (α)$ .

T₁ (T₂) is surjective and T₁ (⋃ _t∈TH_t) = ⋃ _t∈TT₁ (H_t); T₁ (⋂ _t∈TH_t) = ⋂ _t∈TT₁ (H_t); T₁ (H^c) = (T₁ (H)) ^c.

Proof. (1) Let $T_{1} (H) = (F, E) \in VSS (X), H (λ) = (H_{λ}, E) \in FS (3^{X})$ . We first show that $(F, E)_{\underline{λ}} \subseteq H (λ) \subseteq (F, E)_{λ}$ . Since (F, E) = T₁ (H) = ⋃ _λ∈[0,1]λH (λ), for ∀e ∈ E, ∀ x ∈ X, t_F(e) (x) = ⋁ {α|α ∈ [0, 1] , H_α (e) (x) =1} , $1 - f_{F (e)} (x) = ⋁ {α | α \in [0, 1], H_{α} (e) (x) \geq \frac{1}{2}} .$ If H_λ (e) (x) =1, then t_F(e) (x) = ⋁ {α|α ∈ [0, 1] , H_α (e) (x) =1} ≥ λ, soF_λ (e) (x) =1. If F_λ (e) (x) =0, then $λ > 1 - f_{F (e)} (x) = ⋁ {α | α \in [0, 1], H_{α} (e) (x) \geq \frac{1}{2}}$ , soH_λ (e) (x) =0. Because (F, E) _λ, $H (λ) = (H_{λ}, E) \in FS (3^{X})$ , we have H (λ) ⊆ (F, E) _λ. On the other hand, when $F_{\underline{λ}} (e) (x) = 1$ , we have t_F(e) (x) = ⋁ {α|α ∈ [0, 1] , H_α (e) (x) =1} > λ. Thus there exists α > λ such that H_α (e) (x) =1. Therefore, H_λ (e) (x) ≥ H_α (e) (x) =1, i.e. H_λ (e) (x) =1. If H_λ (e) (x) =0, then $H_{α} (e) (x) \geq \frac{1}{2}$ implies that α < λ. Thus $1 - f_{F (e)} (x) = ⋁ {α | α \in [0, 1], H_{α} (e) (x) \geq \frac{1}{2}} \leq ⋁ {α | α \in [0, 1], α < λ} = λ$ . Therefore $F_{\underline{λ}} (e) (x) = 0$ . Because $(F, E)_{\underline{λ}}$ , $H (λ) = (H_{λ}, E) \in FS (3^{X})$ , we have $(F, E)_{\underline{λ}} \subseteq H (λ)$ .

Hence $(F, E)_{\underline{λ}} \subseteq H (λ) \subseteq (F, E)_{λ}$ . By Theorem 4.1, we have T₁ (H) = T₂ (H).

(2) By Theorem 4.1, we know that T₁ (H) _λ = ⋂ _α<λH (α), $T_{1} (H)_{\underline{λ}} = ⋃_{α > λ} H (α)$ .

(3) Let (F, E) ∈ VSS (X), $H (λ) \in U (X)$ and H (λ) = (F, E) _λ. By Theorem 4.1, we have T₁ (H) = (F, E), so T₁ is surjective. Let $H_{t} \in U (X) (t \in T)$ . By Theorem 4.1 and (7) of Property 3.1, we have $T_{1} (⋃_{t \in T} H_{t})_{\underline{λ}} = ⋃_{α > λ} (⋃_{t \in T} H_{t}) (α) = ⋃_{α > λ} ⋃_{t \in T} H_{t} (α) = ⋃_{t \in T} ⋃_{α > λ} H_{t} (α) = ⋃_{t \in T} T_{1} (H_{t})_{\underline{λ}} = (⋃_{t \in T} T_{1} (H_{t}))_{\underline{λ}} .$ $T_{1} (⋂_{t \in T} H_{t})_{λ} = ⋂_{α < λ} (⋂_{t \in T} H_{t}) (α) = ⋂_{α < λ} ⋂_{t \in T} H_{t} (α) = ⋂_{t \in T} ⋂_{α < λ} H_{t} (α) = ⋂_{t \in T} T_{1} (H_{t})_{\underline{λ}} = (⋂_{t \in T} T_{1} (H_{t}))_{\underline{λ}}$ Hence T₁ (⋃ _t∈TH_t) = ⋃ _t∈TT₁ (H_t), T₁ (⋂ _t∈TH_t) = ⋂ _t∈TT₁ (H_t). Similarly, we have $T_{1} (H^{c})_{λ} = ⋂_{α < λ} H^{c} (α) = ⋂_{α < λ} (H (1 - α))^{c} = ⋃_{1 - α > 1 - λ} (H (1 - α))^{c} = (T_{1} (H)_{\underline{1 - λ}})^{c} = ((T_{1} (H))^{c})_{λ} .$ Therefore T₁ (H^c) = (T₁ (H)) ^c. □

Theorem 5.2. Let $T_{i} : U (X) \to VSS (X)$ be a mapping (i = 3, 4), where T₃ (H) = ⋃ _λ∈[0,1]λ ∗ H (λ), T₄ (H) = ⋂ _λ∈[0,1]λ ⋄ H (λ). Then

T₃ (H) = T₄ (H).

$T_{3} (H)^{[λ]} = ⋂_{α < λ} H (α), T_{3} (H)^{\underline{[λ]}} = ⋃_{α > λ} H (α)$ .

T₃ (T₄) is surjective and T₃ (⋃ _t∈TH_t) = ⋂ _t∈TT₃ (H_t); T₃ (⋂ _t∈TH_t) = ⋃ _t∈TT₃ (H_t); T₃ (H^c) = (T₃ (H)) ^c.

Proof. The proof is similar to that of Theorem 5.1. □

Proposition 5.1. T₁ (H) = T₂ (H) = (T₃ (H)) ^c = (T₄ (H)) ^c.

Proof. From Theorem 5.1, we know that T₁ (H) = ⋃ _λ∈[0,1]λH (λ), T₂ (H) = ⋂ _λ∈[0,1]λ · H (λ) and T₁ (H) = T₂ (H). Because (λG, E) = (λ ⋄ G, E) ^c, (λ · G, E) = (λ ∗ G, E) ^c, T₃ (H) = ⋃ _λ∈[0,1]λ ∗ H (λ) = ⋃ _λ∈[0,1] (λ · H (λ)) ^c = (⋂ _λ∈[0,1]λ · H (λ)) ^c = (T₂ (H)) ^c . T₄ (H) = ⋂ _λ∈[0,1]λ ⋄ H (λ) = ⋂ _λ∈[0,1] (λ H (λ)) ^c = (⋃ _λ∈[0,1]λH (λ)) ^c = (T₁ (H)) ^c . Hence T₁ (H) = T₂ (H) = (T₃ (H)) ^c = (T₄ (H)) ^c. □

Definition 5.2. Let $T_{1} : U (X) \to VSS (X)$ be a mapping in Theorem 5.1, $\forall H_{1}, H_{2} \in U (X)$ . Define the relation ∼ in $U (X)$ as H₁ ∼ H₂ ⇔ T₁ (H₁) = T₁ (H₂). Then ∼ is a equivalent relation on $U (X)$ .

Let ${H} = {H_{1} | H_{1} \in U (X), H_{1} \sim H}$ , then factor set $U (X) / \sim = {{H} | H \in U (X)}$ .

Lemma 5.1. Let $H, H^{'}, H_{t}, H_{t}^{'} \in U (X) (t \in T)$ . If ∀t ∈ T, H ∼ H′, $H_{t} \sim H_{t}^{'}$ , then

$(⋃_{t \in T} H_{t}) \sim (⋃_{t \in T} H_{t}^{'})$ .

$(⋂_{t \in T} H_{t}) \sim (⋂_{t \in T} H_{t}^{'})$ .

H^c ∼ (H′) ^c.

Proof. (1) Since $H_{t} \sim H_{t}^{'}$ , then $T_{1} (H_{t}) = T_{1} (H_{t}^{'})$ , so $T_{1} (⋃_{t \in T} H_{t}) = ⋃_{t \in T} T_{1} (H_{t}) = ⋃_{t \in T} T_{1} (H_{t}^{'}) = T_{1} (⋃_{t \in T} H_{t}^{'}) .$ Hence $(⋃_{t \in T} H_{t}) \sim (⋃_{t \in T} H_{t}^{'})$ .

(2) $T_{1} (⋂_{t \in T} H_{t}) = ⋂_{t \in T} T_{1} (H_{t}) = ⋂_{t \in T} T_{1} (H_{t}^{'}) = T_{1} (⋂_{t \in T} H_{t}^{'}) .$ Hence $(⋂_{t \in T} H_{t}) \sim (⋂_{t \in T} H_{t}^{'})$ .

(3) Since H ∼ H′, then T₁ (H) = T₁ (H′), so T₁ (H^c) = (T₁ (H)) ^c = (T₁ (H′)) ^c = T₁ ((H′) ^c) . HenceH^c ∼ (H′) ^c. □

In $U (X) / \sim$ , we define

⋃_t∈T {H_t} : ⋃ _t∈T {H_t} = {⋃ _t∈TH_t}.

⋂_t∈T {H_t} : ⋂ _t∈T {H_t} = {⋂ _t∈TH_t}.

{H} ^c : {H} ^c = {H^c}.

From Lemma 5.1, the operations defined above are not related to the select of representative element.

Theorem 5.3. $U (X) / \sim ≅ VSS (X)$ .

Proof. Let $T^{'} : U (X) / \sim \to VSS (X)$ be a mapping, where T′ ({H}) = T₁ (H).

T′ is bijective.

We show that T′ preserve operations. Let ${H_{t}} \in U (X) / \sim$ , T′ (⋃ _t∈T {H_t}) = T′ ({⋃ _t∈TH_t}) = T₁ (⋃ _t∈TH_t) = ⋃ _t∈TT₁ (H_t) = ⋃ _t∈TT′ ({H_t}) . T′ (⋂ _t∈T {H_t}) = T′ ({⋂ _t∈TH_t}) = T₁ (⋂ _t∈TH_t) = ⋂ _t∈TT₁ (H_t) = ⋂ _t∈TT′ ({H_t}) . T′ ({H} ^c) = T′ ({H^c} = T₁ (H^c) = (T₁ (H)) ^c = (T′ ({H})) ^c . □

Therefore, vague soft sets are the equivalent class of inverse order nested sets in fact.

Definition 5.3. Let $H : [0, 1] \to FS (3^{X})$ be a mapping. If λ₁ < λ₂ ⇒ H (λ₁) ⊆ H (λ₂), then H is called a order nested set on X. We use $V (X)$ to denote the set of all order nested sets of X. In $V (X)$ , we define

H₁ ⊆ H₂ ⇔ H₁ (α) ⊇ H₂ (α) , α ∈ [0, 1].

⋃_t∈TH_t : (⋃ _t∈TH_t) (α) = ⋂ _t∈TH_t (α) , α ∈ [0, 1].

⋂_t∈TH_t : (⋂ _t∈TH_t) (α) = ⋃ _t∈TH_t (α) , α ∈ [0, 1].

H^c : H^c (α) = (H (1 - α)) ^c, α ∈ [0, 1].

Based on Theorems 4.3 and 4.4, Properties 3.3 and 3.4, we can obtain the following two theorems.

Theorem 5.4. Let $T_{i} : V (X) \to VSS (X)$ be a mapping (i = 5, 6), where T₅ (H) = ⋃ _λ∈[0,1]λ ★ H (λ), T₆ (H) = ⋂ _λ∈[0,1]λ ∘ H (λ). Then

T₅ (H) = T₆ (H).

T₅ (H) _[λ] = ⋂ _α>λH (α), $T_{5} (H)_{\underline{[λ]}} = ⋃_{α < λ} H (α)$ .

T₅ (T₆) is surjective and T₅ (⋃ _t∈TH_t) = ⋂ _t∈TT₅ (H_t); T₅ (⋂ _t∈TH_t) = ⋃ _t∈TT₅ (H_t); T₅ (H^c) = (T₅ (H)) ^c.

Theorem 5.5. Let $T_{i} : V (X) \to VSS (X)$ be a mapping (i = 7, 8), where T₇ (H) = ⋃ _λ∈[0,1]λ ⊗ H (λ), T₈ (H) = ⋂ _λ∈[0,1]λ ⊙ H (λ). Then

T₇ (H) = T₈ (H).

$T_{7} (H)^{λ} = ⋂_{α > λ} H (α), T_{7} (H)^{\underline{λ}} = ⋃_{α < λ} H (α)$ .

T₇ (T₈) is surjective and T₇ (⋃ _t∈TH_t) = ⋃ _t∈TT₇ (H_t); T₇ (⋂ _t∈TH_t) = ⋂ _t∈TT₇ (H_t); T₇ (H^c) = (T₇ (H)) ^c.

Proposition 5.2. T₇ (H) = T₈ (H) = (T₅ (H)) ^c = (T₆ (H)) ^c.

Proof. From Theorem 5.4, we know that T₅ (H) = ⋃ _λ∈[0,1]λ ★ H (λ), T₆ (H) = ⋂ _λ∈[0,1]λ ∘ H (λ) andT₅ (H) = T₆ (H). Because (λ ★ G, E) = (λ⊙G, E) ^c, (λ ∘ G, E) = (λ ⊗ G, E) ^c, T₇ (H) = ⋃ _λ∈[0,1]λ ⊗ H (λ) = ⋃ _λ∈[0,1] (λ ∘ H (λ)) ^c = (⋂ _λ∈[0,1]λ ∘ H (λ)) ^c = (T₆ (H)) ^c . T₈ (H) = ⋂ _λ∈[0,1]λ ⊙ H (λ) = ⋂ _λ∈[0,1] (λ ★ H (λ)) ^c = (⋃ _λ∈[0,1]λ ★ H (λ)) ^c = (T₅ (H)) ^c . Hence T₇ (H) = T₈ (H) = (T₅ (H)) ^c = (T₆ (H)) ^c. □

Definition 5.4. Let $T_{7} : V (X) \to VSS (X)$ be a mapping in Theorem 5.5, $\forall H_{1}, H_{2} \in V (X)$ . Define the relation ∼ in $V (X)$ as H₁ ∼ H₂ ⇔ T₇ (H₁) = T₇ (H₂). Then ∼ is a equivalent relation on $V (X)$ .

Let ${H} = {H_{1} | H_{1} \in V (X), H_{1} \sim H}$ , then factor set $V (X) / \sim = {{H} | H \in V (X)}$ .

Lemma 5.2. Let $H, H^{'}, H_{t}, H_{t}^{'} \in V (X) (t \in T)$ . If ∀t ∈ T, H ∼ H′, $H_{t} \sim H_{t}^{'}$ , then

$(⋃_{t \in T} H_{t}) \sim (⋃_{t \in T} H_{t}^{'})$ .

$(⋂_{t \in T} H_{t}) \sim (⋂_{t \in T} H_{t}^{'})$ .

H^c ∼ (H′) ^c.

Proof.

Since $H_{t} \sim H_{t}^{'}$ , then $T_{7} (H_{t}) = T_{7} (H_{t}^{'})$ , so $T_{7} (⋃_{t \in T} H_{t}) = ⋃_{t \in T} T_{7} (H_{t}) = ⋃_{t \in T} T_{7} (H_{t}^{'}) = T_{7} (⋃_{t \in T} H_{t}^{'}) .$ Hence $(⋃_{t \in T} H_{t}) \sim (⋃_{t \in T} H_{t}^{'})$ .

$T_{7} (⋂_{t \in T} H_{t}) = ⋂_{t \in T} T_{7} (H_{t}) = ⋂_{t \in T} T_{7} (H_{t}^{'}) = T_{7} (⋂_{t \in T} H_{t}^{'}) .$ Hence $(⋂_{t \in T} H_{t}) \sim (⋂_{t \in T} H_{t}^{'})$ .

Since H ∼ H′, then T₇ (H) = T₇ (H′), so T₇ (H^c) = (T₇ (H)) ^c = (T₇ (H′)) ^c = T₇ ((H′) ^c) .Hence H^c ∼ (H′) ^c. □

In $V (X) / \sim$ , we define

⋃_t∈T {H_t} : ⋃ _t∈T {H_t} = {⋃ _t∈TH_t}.

⋂_t∈T {H_t} : ⋂ _t∈T {H_t} = {⋂ _t∈TH_t}.

{H} ^c : {H} ^c = {H^c}.

From Lemma 5.2, the operations defined above are not related to the select of representative element.

Theorem 5.6. $V (X) / \sim ≅ VSS (X)$ .

Proof. Let $T^{'} : V (X) / \sim \to VSS (X)$ be a mapping, where T′ ({H}) = T₇ (H).

T′ is bijective.

We show that T′ preserve operations. Let ${H_{t}} \in V (X) / \sim$ , T′ (⋃ _t∈T {H_t}) = T′ ({⋃ _t∈TH_t}) = T₇ (⋃ _t∈TH_t) = ⋃ _t∈TT₇ (H_t) = ⋃ _t∈TT′ ({H_t}) . T′ (⋂ _t∈T {H_t}) = T′ ({⋂ _t∈TH_t}) = T₇ (⋂ _t∈TH_t) = ⋂ _t∈TT₇ (H_t) = ⋂ _t∈TT′ ({H_t}) .T′ ({H} ^c) = T′ ({H^c}) = T₇ (H^c) = (T₇ (H)) ^c = (T′ ({H})) ^c . □

Therefore, vague soft sets are the equivalent class of order nested sets in fact.

6 Conclusion

In this paper, the four kinds of cut sets of vague soft sets are defined by triple valued fuzzy soft sets. The properties of these kinds of cut sets and their relations are discussed. For each kind of cut set, four decomposition theorems are established. Based on decomposition theorems, representation theorems are obtained. These discussion extended the theories of vague soft sets.

Conflict of interests

The author declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

The author would like to thank the anonymous referees for their constructive comments as well as helpful suggestions from the associate editor which helped in improving this paper significantly. The works described in this paper are supported by the National Natural Science Foundation of China under Grant Nos. 11571276, 11501444; the Postdoctoral Science Foundation of China under Grant Nos. 2013M532079, 2014T70932; the Science Research Foundation of Education Department of Shaanxi Provincial Government under Grant No. 15JK1735.

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