**Proof.**


**Proposition 2.** *Let C* F *be a SVN β-covering of U and C* F = {*<sup>C</sup>*1, *C*2, ... , *Cm*}*. Then, for all x* ∈ *U, x* ∈ N*βy if and only if* N*βx* ⊆ N*βy .*

**Proof.** (⇒): For any *z* ∈ N*βx* , we know N*βx* (*z*) ≥ *β*. Since *x* ∈ N*βy* , N*βy* (*x*) ≥ *β*. According to (2) in Theorem 1, we have N *β y* (*z*) ≥ *β*. Hence, *z* ∈ N*βy* . Therefore, N*βx* ⊆ N*βy* . (⇐): According to (1) in Theorem 2, *x* ∈ N*βx* for all *x* ∈ *U*. Since N*βx* ⊆ N*βy* , *x* ∈ N*βy* .

The relationship between SVN *β*-neighborhoods and *β*-neighborhoods is presented in the following proposition.

**Proposition 3.** *Let C* F *be a SVN β-covering of U. For any x*, *y* ∈ *U,* N *β x* ⊆ N *β y if and only if* N*βx* ⊆ N*βy .*

**Proof.** According to Propositions 1 and 2, it is straightforward.

#### **4. Two Types of Single Valued Neutrosophic Covering Rough Set Models**

In this section, we propose two types of SVN covering rough set models on basis of the SVN *β*-neighborhoods and the *β*-neighborhoods, respectively. Then, we investigate the properties of the defined lower and upper approximation operators.

**Definition 6.** *Let* (*<sup>U</sup>*,*<sup>C</sup>* F ) *be a SVN β-covering approximation space. For each A* ∈ *SVN*(*U*) *where A* = {*<sup>x</sup>*, *TA*(*x*), *IA*(*x*), *FA*(*x*) : *x* ∈ *<sup>U</sup>*}*, we define the single valued neutrosophic (SVN) covering upper approximation* C (*A*) *and lower approximation* C (*A*) *of A as:*

$$\begin{split} \bar{\mathsf{C}}(A) &= \{ \langle \mathsf{x}, \vee\_{\mathcal{Y} \in \mathcal{U}} [T\_{\widehat{\mathsf{N}}^{\beta}\_{\mathbbm{x}}}(\mathcal{y}) \wedge T\_{A}(\mathcal{y})], \vee\_{\mathcal{Y} \in \mathcal{U}} [I\_{\widehat{\mathsf{N}}^{\beta}\_{\mathbbm{x}}}(\mathcal{y}) \wedge I\_{A}(\mathcal{y})], \wedge\_{\mathcal{Y} \in \mathcal{U}} [F\_{\widehat{\mathsf{N}}^{\beta}\_{\mathbbm{x}}}(\mathcal{y}) \vee F\_{A}(\mathcal{y})] \}: \mathbf{x} \in \mathsf{U} \}, \\ \mathsf{C}(\mathcal{A}) &= \{ \langle \mathsf{x}, \wedge\_{\mathcal{Y} \in \mathcal{U}} [F\_{\widehat{\mathsf{N}}^{\beta}\_{\mathbbm{x}}}(\mathcal{y}) \vee T\_{A}(\mathcal{y})], \wedge\_{\mathcal{Y} \in \mathcal{U}} [(1 - I\_{\widehat{\mathsf{N}}^{\beta}\_{\mathbbm{x}}}(\mathcal{y})) \vee I\_{A}(\mathcal{y})], \vee\_{\mathcal{Y} \in \mathcal{U}} [T\_{\widehat{\mathsf{N}}^{\beta}\_{\mathbbm{x}}}(\mathcal{y}) \wedge F\_{A}(\mathcal{y})] \}: \mathbf{x} \in \mathsf{U} \}. \end{split} \tag{5}$$

If C (*A*) = C (*A*), then *A* is called the first type of SVN covering rough set.

∼

**Example 3** (Continued from Example 1)**.** *Let β* = 0.5, 0.3, 0.8*, A* = (0.6,0.3,0.5) *x*1 + (0.4,0.5,0.1) *x*2 + (0.3,0.2,0.6) *x*3+ (0.5,0.3,0.4) *x*4+ (0.7,0.2,0.3) *x*5*. Then,*

$$
\widetilde{\mathbb{C}}(A) = \{ \langle \mathbf{x}\_1, 0.6, 0.3, 0.5 \rangle, \langle \mathbf{x}\_2, 0.4, 0.3, 0.6 \rangle, \langle \mathbf{x}\_3, 0.6, 0.5, 0.5 \rangle, \langle \mathbf{x}\_4, 0.5, 0.3, 0.6 \rangle, \langle \mathbf{x}\_5, 0.6, 0.5, 0.5 \rangle \}, \ldots \}
$$

$$
\underline{\mathbb{C}}(A) = \{ \langle \mathbf{x}\_1, 0.6, 0.5, 0.5 \rangle, \langle \mathbf{x}\_2, 0.6, 0.5, 0.4 \rangle, \langle \mathbf{x}\_3, 0.4, 0.4, 0.5 \rangle, \langle \mathbf{x}\_4, 0.4, 0.5, 0.4 \rangle, \langle \mathbf{x}\_5, 0.6, 0.4, 0.3 \rangle \}.
$$

Some basic properties of the SVN covering upper and lower approximation operators are proposed in the following proposition.

**Proposition 4.** *Let C* F *be a SVN β-covering of U. Then, the SVN covering upper and lower approximation operators in Definition 6 satisfy the following properties: for all A*, *B* ∈ *SVN*(*U*)*,*

(1) C (*A* )=(C (*A*)) *,* C (*A* )=(C (*A*)) *.*

∼

∼

∼

∼

∼

∼

∼ (2) *If A* ⊆ *B, then* C (*A*) ⊆ C (*B*)*,* C (*A*) ⊆ C (*B*)*.*

∼

∼ (3) C (*A* G *B*) = C (*A*) G C ∼(*B*)*,* C (*A* A *B*) = C (*A*) A C (*B*)*.*

(4) C (*A* A *B*) ⊇ C (*A*) A C ∼(*B*)*,* C (*A* G *B*) ⊆ C (*A*) G C (*B*)*.*
