The Rough Linear Approximate Space and Soft Linear Space

1 Introduction

In recent years, different methods, such as fuzzy sets, rough sets, and soft sets, are proposed in order to deal with uncertain problems in real life. The rough sets theory, which was proposed by Pawlak [22], is a tool that can deal with granularity in information systems. The rough sets theory may serve as a new mathematical approach to uncertainties, and has attracted the interest of researchers and practitioners in various fields of science and technology. The notion of rough logic was initially proposed by Pawlak [23], and further studies were done in Refs. [4, 16, 17, 30, 32]. Zhang and Zhu proposed a rough logic system in Ref. [30], the schematic system of which was extensional regular double Stone algebras. The theory of truth degree in three-valued pre-rough logic is studied in Refs. [16, 17]. In recent years, the rough set theory has made great progress in theory and practical applications, such as the covering rough set [33], three-way decisions in rough set [28], rough matroids [34], multigranulation rough set [24], etc. Moreover, the rough set theory has been successfully applied to many fields [6, 19], such as pattern recognition, expert systems, medical diagnosis, environmental science, biology, biochemistry, chemistry psychology, conflict analysis, economics, process control, and elsewhere.

Combination research is a good way to do further research. As we have known, rough algebras are a combination research through rough sets and algebraic systems [11, 21, 26, 29, 31]. In this paper, we will establish an interesting connection between linear space and rough sets. We know that linear space is a very important concept in linear algebra, so the connection of rough set to linear space is useful. At present, Kuroki [13] has studied the properties of rough sets in linear space, putting forward the equivalence relation and the upper and lower approximations in linear space. Liu and Yang [14, 15] and Wu [27] studied further properties based on the properties of linear space. They introduced rough sets into linear space and fuzzy linear space, and gave some properties; however, they did not study the algebraic structure of rough linear space. In this paper, we will reveal that rough linear space is a Boolean algebra under the intersection, union, and complementary operators.

Soft set was proposed by Molodtsov in Ref. [20], which is a new mathematical tool to deal with uncertain problems. It overcomes the shortcoming of the traditional fuzzy set theory – lack of parameter sets. At present, soft sets have had a great development in theory and application [1–3, 7–10, 12, 18]. Maji et al. [18] have further studied the theory of soft sets. Feng et al. [7–10] have further studied soft sets and introduced the soft rough set, and they have applied the soft set theoretic approach in decision-making problems, with further applications in Refs. [5, 25]. Ali et al. [2, 3, 12] have further studied the algebraic structures of soft set and fuzzy soft set. We can see that there are no combination studies on soft sets and linear space, so we will build a connection between linear space and soft sets.

The main aim of this paper was to build the connection between linear space and rough (soft) sets. First, the inclusion relation of the upper approximation’s intersection and the inclusion relation of the lower approximation’s union are improved. We obtain the equations of the upper approximation’s intersection and the lower approximation’s union. Next, the rough linear approximate space is proposed, and we prove that the rough linear approximate space is a Boolean algebra on the intersection, union, and complementary operators. In the following, the soft set theory is applied to the linear space, the definitions of soft linear space and soft linear subspace are introduced and some properties are explored, the notion of homomorphism between linear space is introduced, and some basic properties are presented. Lastly, the definitions of lower and upper approximations of soft linear space are given and their properties are studied.

The paper is organized as follows. After this introduction, Section 2 will introduce the definition and properties of rough linear space. In Section 3, two new operators are given in rough linear space. In Section 4, the algebraic structure of rough linear approximation space will be established. In Section 5, the definition and properties of soft linear space will be introduced. In Section 6, the approximation operators in soft linear space will be given, and their properties will be studied. The final section offers the conclusion.

2 The Definition and Properties of Rough Linear Space

Rough sets begin with information systems (or information tables), and as a proposed abstraction. In general, an approximation space is a pair (U, R), where U is a non-empty set (the domain of discourse) and R is an equivalence relation on U.

Let (U, R) be an approximation space; if A⊆U, the lower approximation RA of A is the union of equivalence classes contained in A, while its upper approximation R̅A is the union of equivalence classes intersecting A, i.e.

R_A = ∪{[x]|[x] ⊆ A} = {x|x ∈ U, [x] ⊆ A},R¯A = ∪{[x]|[x]∩X≠∅} = {x|x∈U, [x] ∩ A ≠ ∅},

where [x] denotes the equivalence classes containing x. Then, we call R̅A(RA) the rough upper (lower) approximation of A. A rough set A is termed definable in (U, R) if and only if RA(R̅A).

Definition 2.1 ([14]): Let V be a linear space over the number field P; X and Y are non-empty subsets of V, k∈P; the sum of X and Y is defined as

X + Y = {α = α1 + α2|α1 ∈ X, α2 ∈ Y}.

The product of A and number k is defined as kX={kα|α∈X}.

Definition 2.2 ([14]): Let V be a linear space and let W be a subspace of V; α, β∈V are called congruence with respect to W, if α−β∈W, denoted by αρ_Wβ. We can classify the vectors of V according to ρ_W: α and β are in the same ρ_W class iff α−β∈W. We denote by ρ_W(α), the ρ_W-congruence class containing the vector α∈V.

Remark 1: Obviously, ρ_W is an equivalence relation over V, and if α, β∈V, k≠0(k∈P), then ρ_W(α)+ρ_W(β)=ρ_W(α+β), ρ_W(kα)=kρ_W(α).

Definition 2.3 ([14]): Let W be a subspace of a linear space V, if X is a non-empty subset of V, the sets ρ_W(X)={α∈V|ρ_W(α)⊆X} and ρ̅_W(X)={α∈V|ρ_W(α)∩X≠∅} are called the ρ_W-lower and ρ_W-upper approximations of X with respect to W, respectively.

Theorem 2.1 ([14]): Let W be a subspace of a linear space V; if X and Y are non-empty subsets of V, then

ρ_W(X) ⊆ X ⊆ ρ¯W(X),ρ¯W(X ∪ Y) = ρ¯W(X) ∪ ρ¯W(Y),ρ_W(X ∩ Y) = ρ_W(X) ∩ ρ_W(Y).

If X⊆Y, then ρ_W(X)⊆ρ_W(Y),

ρ_W(X ∪ Y) ⊇ ρ_W(X) ∪ ρ_W(Y),ρ¯W(X ∩ Y) ⊆ ρ¯W(X) ∩ ρ¯W(Y).

3 Two New Operators in Linear Space

From Theorem 2.1, we can see that the upper approximation’s intersection and the lower approximation’s union are inclusion relations, and they are not equalities. In this section, we will give their equal expressions.

Definition 3.1: Let W be a subspace of a linear space V; if X and Y are non-empty subsets of V, we define

PX(Y) = {α|ρW(α) ⊆ X ∪ Y,  ρW(α) ⊂ X,  ρW(α) ⊂ Y},QX(Y) = {α|ρW(α) ∩ (X ∩ Y) = ∅,  ρW(α) ∩ X ≠ ∅,  ρW(α) ∩ Y ≠ ∅}.

Theorem 3.1:Let W be a subspace of a linear space V; if X and Y are non-empty subsets of V, then

1. ρ_W(X ∪ Y)=ρ_W(X) ∪ ρ_W(Y) ∪ PX(Y).

2. ρ¯W(X ∩ Y) = ρ¯W(X) ∩ ρ¯W(Y) − QX(Y).

Proof: 1. For any α∈ρ_W(X∪Y), we have ρ_W(α)⊆X∪Y. If ρ_W(α)⊆X or ρ_W(α)⊆Y, then α∈ρ_W(X) or α∈ρ_W(Y), so α∈ρ_W(X)∪ρ_W(Y)∪P_X(Y).

If ρ_W(α)⊄X and ρ_W(α)⊄Y, then α∈P_X(Y), so α∈ρ_W(X)∪ρ_W(Y)∪P_X(Y). Thus, we have ρ_W(X∪Y)⊆ ρ_W(X)∪ρ_W(Y)∪P_X(Y). For any α∈ρ_W(X)∪ρ_W(Y)∪P_X(Y), we have α∈ρ_W(X) or α∈ρ_W(Y). If α∈ρ_W(X), then α∈ρ_W(X∪Y). If α∈ρ_W(Y), then α∈ρ_W(X∪Y). If α∈P_X(Y), then ρ_W(α)⊆X∪Y, so α∈ρ_W(X∪Y). Thus, we have ρ_W(X)∪ρ_W(Y)∪P_X(Y)⊆ρ_W(X∪Y). Hence, ρ_W(X∪Y)=ρ_W(X)∪ρ_W(Y)∪P_X(Y).

2. This proposition is equivalent to ρ̅_W(X∩Y)∪Q_X(Y)=ρ̅_W(X)∩ ρ̅_W(Y). For any α∈ρ̅_W(X∩Y)∪Q_X(Y), we have α∈ρ̅_W(X∩Y) or α∈Q_X(Y). If α∈ρ̅_W(X∩Y), then α∈ρ̅_W(X)∩ρ̅_W(Y). If α∈Q_X(Y), then ρ_W(α)∩X≠∅ and ρ_W(α)∩Y≠∅, so we have α∈ρ̅_W(X)∩ρ̅_W(Y).

For any α∈ρ̅_W(X)∩ρ̅_W(Y), we have α∈ρ̅_W(X) and α∈ρ̅_W(Y); thus, we get ρ_W(α)∩X≠∅ and ρ_W(α)∩Y≠∅.

If ρW(α) ∩ (X ∩ Y) ≠ ∅, then α ∈ ρ¯W(X ∩ Y).

If ρ_W(α)∩(X∩Y)=∅, then α∈Q_X(Y). Thus, we have α∈ρ̅_W(X∩Y)∩Q_X(Y). Hence, ρ̅_W(X∩Y)=ρ̅_W(X)∩ρ̅_W (Y)−Q_X(Y).         □

Example 3.1: If α is a vector and span (α) represents the generated subspace of α. R⁴ represents the four-dimensional space on real set R. Let W=span((1,0,0,0), (0,1,0,0)), X=span((1,0,0,0,0), (0,1,0,0), (0,0,1,0)), and Y=span((1,0,0,0,0), (0,0,1,0), (0,0,0,1)).

Then, ρ_W(X)=span((1,0,0,0,0), (0,1,0,0), (0,0,1,0)), ρ_W(Y)=∅, ρ_W(X∪Y)=R⁴, ρ̅_W(X)=R⁴, ρ̅_W(Y)=R⁴, and ρ̅_W(X∩Y)=R⁴. P_X(Y)=span((0,0,0,0)), Q_X(Y)=∅. Thus, we have ρ_W(X∪Y)=ρ_W(X)∪ρ_W(Y)∪P_X(Y), ρ̅_W(X∩Y)=ρ̅_W(X)∩ρ̅_W(Y)−Q_X(Y).

4 Rough Linear Approximation Space and Its Algebra

In this section, the rough linear approximate space will be proposed, and the intersection, union, and complementary operators will be introduced in rough linear approximate space. Moreover, we will prove that the rough linear approximate space is a Boolean algebra on the intersection, union, and complementary operators.

Definition 4.1: Let V be a linear space and X is a non-empty subset of V; we define

ρW(X) = (ρ_W(X), ρ¯W(X)).

Definition 4.2: Let V be a linear space and X, Y are non-empty subsets of V. The union, intersection, complement, and difference of rough sets are defined as follows:

1. ρW(X) ∪ ρW(Y) = (ρ_W(X) ∪ ρ_W(Y) ∪ PX(Y),  ρ¯W(X) ∪ ρ¯W(Y)).

2. ρW(X) ∩ ρW(Y) = (ρ_W(X) ∩ ρ_W(Y),  ρ¯W(X) ∩ ρ¯W(Y) − QX(Y)).

3. ρW(X⊥) = ρW(X)⊥ = (V − ρ¯W(X),  V − ρ_W(X)).

4. ρW(X) − ρW(Y) = (ρ_W(X) − ρ¯W(Y), ρ¯W(X) − ρ_W(Y) − QX(Y⊥)).

Theorem 4.1:Let V be a linear space and X be a non-empty subset of V. Then, the following holds:

1. ρW(X) ∪ ρW(Y) = ρW(X ∪ Y).

2. ρW(X) ∩ ρW(Y) = ρW(X ∩ Y).

Proof: 1. According to Definition 4.1, we can easily get ρ_W(X∪Y)=(ρ_W(X∪Y), ρ̅_W(X∪Y)).

From Theorem 3.1, we get ρ_W(X∪Y)=ρ_W(X)∪ρ_W(Y)∪P_X(Y), ρ̅_W(X∩Y)=ρ̅_W(X)∩ρ̅_W(Y)−Q_X(Y). Thus, ρ_W(X)∪ρ_W(Y)=ρ_W(X∪Y). Similarly, we can get ρ_W(X)∩ρ_W(Y)=ρ_W(X∩Y).        □

Theorem 4.2:Let V be a linear space, and X, Y and Z are non-empty subsets of V. We have

1. Commutative law:

   ρW(X) ∪ ρW(Y) = ρW(Y) ∪ ρW(X),ρW(X) ∩ ρW(Y) = ρW(Y) ∩ ρW(X).

2. Associative law:

   (ρW(X) ∪ ρW(Y)) ∪ ρW(Z) = ρW(X) ∪ (ρW(Y) ∪ ρW(Z)),(ρW(X) ∩ ρW(Y)) ∩ ρW(Z) = ρW(X) ∩ (ρW(Y) ∩ ρW(Z)).

3. Distributive law:

   ρW(X) ∪ (ρW(Y) ∩ ρW(Z)) = (ρW(X) ∪ ρW(Y)) ∩ (ρW(X) ∪ ρW(Z)),ρW(X) ∩ (ρW(Y) ∪ ρW(Z)) = (ρW(X) ∩ ρW(Y)) ∪ (ρW(X) ∩ ρW(Z)).

4. Idempotent law:

   ρW(X) ∪ ρW(X) = ρW(X),ρW(X) ∩ ρW(X) = ρW(X).

5. 0−1 law:

   ρW(X) ∪ ρW(∅) = ρW(X),ρW(X) ∩ ρW(V) = ρW(X).

6. Complementary law:

   ρW(X) ∪ ρW(X⊥) = ρW(V),ρW(X) ∩ ρW(X⊥) = ρW(∅).

7. De Morgan law:

   (ρW(X) ∪ ρW(Y))⊥ = ρW(X⊥) ∩ ρW(Y⊥),(ρW(X) ∩ ρW(Y))⊥ = ρW(X⊥) ∪ ρW(Y⊥).

Proof:

1. From Definition 3.1, we have P_X(Y)=P_Y(X), Q_X(Y)=Q_Y(X). Thus,

   ρW(X) ∪ ρW(Y) = (ρ_W(X) ∪ ρ_W(Y)∪PX(Y),  ρ¯W(X) ∪ ρ¯W(Y))=(ρ_W(Y) ∪ ρ_W(X) ∪ PY(X), ρ¯W(Y) ∪ ρ¯W(X)) = ρW(Y)∪ρW(X),ρW(X) ∩ ρW(Y) = (ρ_W(X) ∩ ρ_W(Y), ρ¯W(X) ∩ ρ¯W(Y) − QX(Y))=(ρ_W(Y) ∩ ρ_W(X), ρ¯W(Y) ∩ ρ¯W(X) − QY(X)) = ρW(Y) ∩ ρW(X).

2. (ρW(X) ∪ ρW(Y)) ∪ ρW(Z) = ρW(X ∪ Y) ∪ ρW(Z)=ρW((X ∪ Y) ∪ Z)=(ρ_W((X ∪ Y) ∪ Z), ρ¯W((X ∪ Y) ∪ Z))=(ρ_W(X ∪ (Y ∪ Z)), ρ¯W(X ∪ (Y ∪ Z)))=(ρ_W(X) ∪ ρ_W(Y ∪ Z) ∪ PX(Y ∪ Z), ρ¯W(X) ∪ ρ¯W(Y ∪ Z))=((ρW(X) ∪ (ρW(Y) ∪ ρW(Z))), (ρW(X) ∩ ρW(Y)) ∩ ρW(Z))=ρW(X ∩ Y) ∩ ρW(Z) = ρW((X ∩ Y) ∩ Z)=(ρ_W((X ∩ Y) ∩ Z), ρ¯W((X ∩ Y) ∩ Z))=(ρ_W(X ∩ (Y ∩ Z)), ρ¯W(X ∩ (Y ∩ Z)))=(ρ_W(X) ∩ ρ_W(Y ∩ Z), ρ¯W(X) ∩ ρ¯W(Y ∩ Z) − QX(Y ∩ Z))=ρW(X) ∩ (ρW(Y) ∩ ρW(Z)).

3. ρW(X) ∪ (ρW(Y) ∩ ρW(Z))= ρW(X) ∪ ρW(Y ∩ Z)= ρW((X) ∪ (Y ∩ Z)) = (ρ_W(X ∪ (Y ∩ Z)), ρ¯W(X ∪ (Y ∩ Z)))= (ρ_W((X ∪ Y) ∩ (X ∪ Z)), ρ¯W((X ∪ Y) ∩ (X ∪ Z))),= (ρ_W(X ∪ Y) ∩ ρ_W(X ∪ Z), ρ¯W(X ∪ Y) ∩ ρ¯W(X ∪ Z) − QX + Y(X ∪ Z))= ρW(X ∪ Y) ∩ ρW(X ∪ Z)= (ρW(X) ∪ ρW(Y)) ∩ (ρW(X) ∪ ρW(Z))ρW(X) ∩ (ρW(Y) ∪ ρW(Z))= ρW(X) ∩ ρW(Y ∪ Z) = ρW(X) ∩ (Y ∪ Z)= (ρ_W(X ∩ (Y ∪ Z)), ρ¯W(X ∩ (Y ∪ Z)))= (ρ_W((X ∩ Y) ∪ (X ∩ Z)), ρ¯W((X ∩ Y) ∪ (X ∩ Z)))= (ρ_W(X ∩ Y) ∪ ρ_W(X ∩ Z) ∪ PX ∩ Y(X ∩ Z), ρ¯W(X ∩ Y) ∪ ρ¯W(X ∩ Z))= ρW(X ∩ Y) ∪ ρW(X ∩ Z)= (ρW(X) ∩ ρW(Y)) ∪ (ρW(X) ∩ ρW(Z)).

4. ρW(X) ∪ ρW(X) = ρW(X ∪ X) = ρW(X),ρW(X) ∩ ρW(X) = ρW(X ∩ X) = ρW(X).

5. ρW(X) + ρW(∅) = ρW(X + ∅) = ρW(X),ρW(X) ∩ ρW(V) = ρW(X ∩ V) = ρW(X).

6. ρW(X) ∪ ρW(X⊥) = ρW(X ∪ X⊥) = ρW(V),ρW(X) ∩ ρW(X⊥) = ρW(X ∩ X⊥) = ρW(∅).

7. If we want to prove that (ρ_W(X)∪ρ_W(Y))^⊥=ρ_W(X^⊥)∩ρ_W(Y^⊥), we only need to prove (ρ_W(X)∪ρ_W(Y))∩ (ρ_W(X^⊥)∩ρ_W(Y^⊥))=ρ_W(∅), and (ρ_W(X)∪ρ_W(Y))∪(ρ_W(X^⊥)∩ρ_W(Y^⊥))=ρ_W(V)

   (ρW(X) ∪ ρW(Y)) ∩ (ρW(X⊥) ∩ ρW(Y⊥))= [(ρW(X)∪ρW(Y)) ∩ ρW(X⊥)] ∩ ρW(Y⊥)= [(ρW(X) ∩ ρW(X⊥)) ∪ (ρW(Y) ∩ ρW(X⊥))] ∩ ρW(Y⊥)= [∅∪(ρW(Y) ∩ ρW(X⊥))] ∩ ρW(Y⊥)= ρW(Y) ∩ ρW(X⊥) ∩ ρW(Y⊥) = ρW(∅) ∩ R(X⊥) = R(∅).

   (ρW(X) ∪ ρW(Y)) ∪ (ρW(X⊥) ∩ ρW(Y⊥))=(ρW(X) ∪ ρW(Y) ∪ ρW(X⊥)) ∩ (ρW(X) ∪ ρW(Y) ∪ ρW(Y⊥))=(ρW(V) ∪ ρW(Y)) ∩ (ρW(V) ∪ ρW(X)) = ρW(V) ∩ ρW(V) = ρW(V).

Thus, (ρ_W(X)∪ρ_W(Y))^⊥=ρ_W(X^⊥)∩ρ_W(Y^⊥) is proved. In the same way, we can also prove (ρ_W(X)∩ρ_W(Y))^⊥= ρ_W(X^⊥)∪ρ_W(Y^⊥).           □

Definition 4.3: Let V be a linear space; let (V, ρ_W) be a rough linear space. If X⊆V, ρ_W(X)=(ρ_W(X), ρ_W(X)) is called a rough set on (V, ρ_W). F={(V, ρ_W(X))|X⊆V} is called a rough linear approximation space. The union and the intersection partition the complement as follows:

(X, ρW(X)) ∪ (Y, ρW(Y)) = (X∪Y, ρW(X ∪ Y)),(X, ρW(X)) ∩ (Y, ρW(Y)) = (X ∩ Y, ρW(X ∩ Y)),(X, ρW(X))⊥ = (X⊥, ρW(X⊥)),0 = (ρW(∅), ρW(∅)), 1 = (ρW(V), ρW(V)).

Remark 4.1: From Theorem 4.2, we can get that the algebraic system 〈F, ∪, ∩, ⊥, 0, 1〉 is a Boolean algebra.

5 The Definition and Properties of Soft Linear Space

In this section, according to the definition of soft set in the literature [1–3, 7–10, 12, 18, 20], we will propose the soft linear space. Let V be a linear space over number field P and let E be a set of parameters. The power set of V is denoted by 2^V, and A is a subset of E.

Definition 5.1: A pair (F, A) is called a soft set over V if F is a mapping from A into 2^V. We call (F, A) as a soft set over V.

Example 5.1: Let V=span(ε₁, ε₂, ε₃, ε₄), ε₁, ε₂, ε₃, and ε₄ be vectors in V. A={a₁, a₂, a₃}. Define F(a₁)=span(ε₁), and F(a₂)=span(ε₂, ε₃), F(a₃)=span(ε₁, ε₃, ε₄). Thus, the pair (F, A) is called a soft set over V.

Definition 5.2: Let (F, A) and (G, B) be soft sets over V. The extended intersection of (F, A) and (G, B) over V is the soft set (H, C), where C=A∪B, and ∀e∈C,

H(e)={F(e), e ∈ A − BG(e), e ∈ B − AF(e) ∩ G(e), e ∈ A ∩ B.

We write (F, A)∩_ε(G, B)=(H, C).

Definition 5.3: Let (F, A) and (G, B) be soft sets over V. The extended sum of (F, A) and (G, B) over V is the soft set (H, C), where C=A∪B, and ∀e∈C,

H(e) = {F(e), e ∈ A − BG(e), e ∈ B − AF(e) + G(e), e ∈ A ∩ B.

We write (F, A) ∪˜ (G, B) = (H, C).

Example 5.2: Let E be a set of parameters and A={a₁, a₂, a₃}, B={a₁, a₃, a₄} be subsets of E. Let (F, A) and (G, B) be two soft sets over the linear space V=span(ε₁, ε₂, ε₃, ε₄, ε₅) such that F(a₁)=L(ε₁, ε₂), F(a₂)=span(ε₂, ε₃), F(a₃)=span(ε₁, ε₄, ε₅); G(a₁)=span(ε₁), G(a₃)=span(ε₃, ε₄), G(a₄)=span(ε₂, ε₄). Let (F, A) ∩ε (G, B) = (H1, C1), (F, A) ∪˜ (G, B) = (H2, C2). Then,

H1(a1) = F(a1) ∩ G(a1) = L(ε1),H1(a2) = F(a2) = L(ε2, ε3),H1(a3) = F(a3) ∩ G(a3) = L(ε4),H1(a4) = G(a4) = L(ε2, ε4),H2(a1) = F(a1) + G(a1) = L(ε1, ε2),H2(a2) = F(a2) = L(ε2, ε3),H2(a3) = F(a3) + G(a3) = L(ε1, ε3, ε4, ε5),H2(a4) = G(a4) = L(ε2, ε4).

Definition 5.4: Let (F, A) and (G, B) be soft sets over V. The restricted intersection of (F, A) and (G, B) over V is the soft set (H, C), where C=A∩B, and ∀e∈C, H(e)=F(e)∩G(e). If A∩B=∅, we shall denote by ∅_∅ the unique soft set over V.

Definition 5.5: Let (F, A) and (G, B) be soft sets over V. The restricted sum of (F, A) and (G, B) over V is the soft set (H, C), where C=A∩B, and ∀e∈C, H(e)=F(e)+G(e). If A∩B=∅, we shall denote by ∅_∅ the unique soft set over V.

Definition 5.6: A soft set (F, A) over V is said to be a null soft set denoted by ∅_A, if ∀e∈A, F(e)=∅.

Definition 5.7: A soft set (F, A) over V is said to be an absolute soft set denoted by V_A, if ∀e∈A, F(e)=V.

Definition 5.8: A soft set (F, A) over V is said to be a soft linear space over V, if ∀e∈A, F(e) is a linear subspace over V.

Example 5.3: The pair (F, A) in Example 5.1 is also a soft linear space over V.

Definition 5.9: Let (F, A) and (G, B) be soft linear spaces over V. We call (F, A) a soft linear subspace of (G, B), if it satisfies (i) A⊆B and (ii) ∀x∈A, F(x) is a linear subspace over G(x). We denote (F, A)<˜(G, B).

Theorem 5.1:Let (F, A) and (G, B) be soft linear spaces over V; the extended intersection of (F, A) and (G, B) is a soft linear space over V.

Proof: Let (H, C)=(F, A)∩_ε(G, B), ∀x∈C; if x∈A−B, then H(x)=F(x). Because (F, A) is a soft linear space over V, so x∈A, F(x) is a linear subspace over V. H(x) is a linear subspace over V.

Similarly, if x∈B−A, H(x) is a linear subspace over V. If x∈A∩B, H(x)=F(x)∩G(x), because F(x) and G(x) are linear subspaces over V. Thus, H(x) is a linear subspace over V. (H, C) is a soft linear space over V.        □

Theorem 5.2:Let (F, A) and (G, B) be soft linear spaces over V; the extended sum of (F, A) and (G, B) is a soft linear space over V.

Proof: Let (H, C) = (F, A) ∪˜ (G, B), ∀x∈C; if x∈A−B, then H(x)=F(x). Because (F, A) is a soft linear space over V, so x∈A, F(x) is a linear subspace over V. H(x) is a linear subspace over V.

Similarly, if x∈B−A, H(x) is a linear subspace over V. If x∈A∩B, H(x)=F(x)+G(x), because F(x) and G(x) are linear spaces over V. Thus, H(x) is a linear subspace over V. (H, C) is a soft linear space over V.       □

We can also obtain the following theorems:

Theorem 5.3:Let (F, A) and (G, B) be soft linear spaces over V, A∩B≠∅; the restricted intersection of (F, A) and (G, B) is a soft linear space over V.

Theorem 5.4:Let (F, A) and (G, B) be soft linear spaces over V, A∩B≠∅; the restricted sum of (F, A) and (G, B) is a soft linear space over V.

Theorem 5.5:Let (F, A) be a linear space over V. (G₁, B₁) and (G₂, B₂) are soft linear subspaces of (F, A). Then

1. (G1, B1) ∩ε (G2, B2) <˜ (F, A),

2. if B1 ∩ B2 = ∅, then (G1, B1) ∪˜ (G2, B2) <˜ (F, A).

Proof:

1. Let (H, C)=(G₁, B₁)∩_ε(G₂, B₂), because (G₁, B₁) and (G₂, B₂) are linear subspaces of (F, A), then B₁⊆A, B₂⊆A, so C=B₁∪B₂⊆A. ∀x∈C, if x∈B₁−B₂, then H(x)=G₁(x)⊆F(x); if x∈B₂−B₁, then H(x)=G₂(x)⊆F(x); if x∈B₁∩B₂, then H(x)=G₁(x)∩G₁(x)⊆F(x). Thus, (G1, B1) ∩ε (G2, B2) <˜ (F, A).

2. Let (H, C) = (G1, B1) ∪˜ (G2, B2), because (G₁, B₁) and (G₂, B₂) are linear subspaces of (F, A); then, B₁⊆A, B₂⊆A, so C=B₁∪B₂⊆A. Because B₁∩B₂=∅, then ∀x∈C; if x∈B₁−B₂, then H(x)=G₁(x)⊆F(x); if x∈B₂−B₁, then H(x)=G₂(x)⊆F(x). Thus, (G1, B1) ∪˜ (G2, B2) <˜ (F, A).           □

Definition 5.10: Let V₁ and V₂ be linear spaces over field P. f is a homomorphic mapping from V₁ into V₂, which satisfies

1. f(α + β) = f(α) + f(β),

2. f(kα) = kf(α),

where ∀α, β∈V₁. Then, {x∈X|f(x)=0} is called the kernel of f, denoted by ker(f).

Example 5.4: Let V₁=span((1,0,0,0)), V₂=span((0,1,0,0)). We define f as a mapping from V₁ into V₂: ∀c∈R, f((c,0,0,0))=(0,2c,0,0). Then, f is a homomorphic mapping from V₁ to V2.

Theorem 5.6:Let V be a linear space over field P, and X, Y be linear spaces over V. f: X→Y is a homomorphic mapping. If (F, A) is a soft linear space over X, then (f(F), A) is a soft linear space over Y, where f(F): A→2^Y: ∀x∈A, f(F)(x)=f(F(x)).

Proof: Because (F, A) is a soft linear space over X, so ∀x∈A, F(x)⊆X. f is a homomorphic mapping from X into Y, so f(F(x))⊆Y. Because f(F)(x)=f(F(x)), so f(F)(x)⊆Y. Thus, (f(F), A) is a soft linear space over Y.            □

Theorem 5.7:Let f: X→Y is a homomorphic mapping. (F, A) is a soft linear space over X.

1. If ∀x∈A, F(x)=ker(f), then (f(F), A) is a null soft linear space over Y.

2. If f is a surjection, then (F, A) is an absolute soft set over X. Then, (f(F), A) is an absolute soft linear space over Y.

Proof:

1. ∀x ∈A, F(x)=ker(f)={α∈X|f(α)=0}. Because α∈X, then f(α)∈Y. f(F)(x)=f(F(x))=f(α)=0. Thus, (f(F), A) is an absolute soft linear space over Y.

2. (F, A) is an absolute soft linear space over X. ∀x∈A, F(x)=X. Because f is a surjection, then f(X)=Y. Thus, f(F)(x)=f(F(x))=f(X)=Y. Hence, (f(F), A) is an absolute soft linear space over Y            □.

Theorem 5.8:Let f: X→Y be a homomorphic mapping V. If (F, A) and (G, B) are soft linear spaces over X, and(F, A) <˜ (G, B),then (f(F), A) and (f(G), B) are soft linear spaces over Y, and(f(F), A) <˜ (f(G), B).

Proof: Because f is a homomorphic mapping from X into Y, by Theorem 5.6, we can get that (f(F), A) and (f(G), B) are soft linear spaces over Y. Because (F, A) <˜ (G, B), then A⊆B, ∀x∈A, F(x)⊆G(x). Thus, f(F(x))⊆f(G(x)), f(F)(x)⊆f(G)(x). Hence, (f(F), A) <˜ (f(G), B).           □

6 The Approximation Operator in Soft Linear Space

In this section, the lower and upper approximations of X in soft linear space will be built and the properties will be studied.

Definition 6.1: Let (F, A) be a soft set over V. Let X be a subspace.

Apr_Q(X)={a∈A,F(a)⊆X∧F(a)≠{0}} andApr¯Q(X)={a∈A,F(a)∩X≠{0}∨F(a)={0}}

are called the lower and upper approximations of X in the soft linear space (V, F, A), respectively.

We will use pos_Q(X)=Apr_Q(X) to denote the positive region of X, negQ(X) = A − Apr¯Q(X) to denote the negative region of X, and bnQ(X) = Apr¯Q(X) − Apr_Q(X) to denote the borderline region of X.

Theorem 6.1:Let (F, A) be a soft set over V and let X, Y be two subspaces of V; we have

1. Apr_Q(X) ⊆ Apr¯Q(X).

2. Apr_Q(X ∩ Y) = Apr_Q(X) ∩ Apr_Q(Y).

3. Apr¯Q(X + Y) = Apr¯Q(X) ∪ Apr¯Q(Y).

4. X ⊆ Y ⇒ Apr_Q(X) ⊆ Apr_Q(Y), Apr¯Q(X) ⊆ Apr¯Q(Y).

5. Apr¯Q(X ∩ Y) ⊆ Apr¯Q(X) ∩ Apr¯Q(Y).

6. Apr_Q(X + Y) ⊇ Apr_Q(X) ∪ Apr_Q(Y).

7. Apr_Q(X) = ∼Apr¯Q(∼X), Apr¯Q(X)=∼Apr_Q(∼X).

Proof:

1. ∀x ∈Apr_Q(X), we get x∈A, F(x)⊆X and F(x)≠{0}. Thus, F(x)∩X≠{0}. Hence, x ∈ Apr¯Q(X), i.e. Apr_Q(X) ⊆ Apr¯Q(X).

2. ∀x ∈ Apr_Q(X ∩ Y) ⇔ x ∈ A, F(x) ⊆ X ∩ Y ∧ F(x) ≠ {0} ⇔ F(x) ⊆ X ∧ F(x) ⊆ Y ∧ F(x) ≠ {0} ⇔ x ∈ Apr_Q(X) ∧ x ∈ Apr_Q(Y) ⇔ x ∈ Apr_Q(X) ∩ Apr_Q(Y), so Apr_Q(X ∩ Y) = Apr_Q(X) ∩ Apr_Q(Y).

3. ∀x ∈ Apr¯Q(X + Y) ⇔ x ∈ A, F(x) ∩ (X + Y) ≠ {0} ∨ F(x) = {0} ⇔ F(x) ∩ X ≠ {0} ∨ F(x) ∩ Y ≠ {0} ∨ F(x) = {0} ⇔ x ∈ Apr¯Q(X) ∨ x ∈ Apr¯Q(Y) ⇔ x ∈ Apr¯Q(X) ∪ Apr¯Q(Y), so Apr¯Q(X + Y) = Apr¯Q(X) ∪ Apr¯Q(Y).

4. ∀x∈Apr_Q(X), we get x∈A, F(x)⊆X, and F(x)≠{0}. From X⊆Y, we get F(x)⊆Y. Thus, x∈Apr_Q(Y). Hence, Apr_Q(X)⊆Apr_Q(Y).

   From ∀x ∈ Apr¯Q(X), we get x∈A, F(x)∩X≠{0}∨F(x)={0}. From X⊆Y, we get F(x)∩Y≠{0}. Thus, F(x)∩Y≠{0}. Hence, Apr¯Q(X) ⊆ Apr¯Q(Y).

5. From X∩Y⊆X, we get Apr¯Q(X ∩ Y) ⊆ Apr¯Q(X). From X∩Y⊆Y, we get Apr¯Q(X ∩ Y) ⊆ Apr¯Q(Y). Thus, Apr¯Q(X ∩ Y) ⊆ Apr¯Q(X) ∩ Apr¯Q(Y).

6. From X⊆X+Y, we get Apr_Q(X)⊆Apr_Q(X+Y). From Y⊆X+Y, we get Apr_Q(Y)⊆Apr_Q(X+Y). Thus, Apr_Q(X+Y)⊇Apr_Q(X)∪Apr_Q(Y).

7. ∀x ∈ Apr_Q(X) ⇔ x ∈ A, F(x) ⊆ X ∧ F(x) ≠ {0} ⇔ x ∈ A, F(x) ∩ (∼X) = {0} ∧ F(x) ≠ {0} ⇔ x ∉ Apr¯Q(∼X) ⇔ x ∈ ∼Apr¯Q(∼X). Thus, Apr_Q(X) = ∼Apr¯Q(∼X).

8. Replace ~X with X, we can get Apr¯Q(X) = ∼Apr_Q(∼X).