**Proof.**


**Proposition 1.** *Let C* F *be a SVN β-covering of U. For any x*, *y* ∈ *U,* N *β x* (*y*) ≥ *β if and only if* N *β y* ⊆ N *β x .*

**Proof.** Suppose the SVN number *β* = *a*, *b*, *c*. (⇒): Since N *β x* (*y*) ≥ *β*,

$$T\_{\widetilde{\mathcal{N}}\_{\mathbf{z}}^{\beta}}(y) = T\_{\bigcap\_{\substack{\mathcal{T}\_{\mathbf{C}\_{i}}(\mathbf{x}) \geq \mathbf{z} \\ \mathcal{I}\_{\mathbf{C}\_{i}}(\mathbf{x}) \leq b \\ \mathcal{F}\_{\mathbf{C}\_{i}}(\mathbf{x}) \leq \varepsilon}} \mathsf{C}\_{\mathbf{f}}(y) = \bigwedge\_{\substack{\mathcal{T}\_{\mathbf{C}\_{i}}(\mathbf{x}) \geq \mathbf{z} \\ \mathcal{I}\_{\mathbf{C}\_{i}}(\mathbf{x}) \leq b \\ \mathcal{F}\_{\mathbf{C}\_{i}}(\mathbf{x}) \leq \varepsilon}} T\_{\mathbf{C}\_{i}}(y) \geq a \,\ \mathsf{Y}\_{\widetilde{\mathcal{N}}\_{\mathbf{z}}^{\beta}}(y) = \operatorname{I}\_{\mathbf{C}\_{i}}(\mathbf{y}) = \operatorname{I}\_{\mathbf{C}\_{i}}(\mathbf{y}) \leq b \,\ \mathsf{Y}\_{\widetilde{\mathcal{N}}\_{\mathbf{z}}^{\beta}}(\mathbf{x}) \geq a$$

and

$$F\_{\overline{\mathbb{M}}\_x^{\emptyset}}(y) = F\_{\bigcap\_{\substack{\mathbb{C}\_{\overline{\mathbb{C}}\_i}(x) \ge a \\ l\_{\mathbb{C}\_i}(x) \le c \\ F\_{\overline{C}\_i}(x) \le b \\ F\_{\overline{C}\_i}(x) \le b}}(y) = \bigvee\_{\substack{T\_{\overline{\mathbb{C}}\_i}(x) \ge a \\ l\_{\mathbb{C}\_i}(x) \le b \\ F\_{\overline{C}\_i}(x) \le c}} F\_{\overline{C}\_i}(y) \le c.$$

Then,

$$\{\mathbb{C}\_{i}\in\hat{\mathbf{C}}:T\_{\mathbb{C}\_{i}}(\mathbf{x})\geq a, I\_{\mathbb{C}\_{i}}(\mathbf{x})\leq b, F\_{\mathbb{C}\_{i}}(\mathbf{x})\leq c\}\subseteq\{\mathbb{C}\_{i}\in\hat{\mathbf{C}}:T\_{\mathbb{C}\_{i}}(y)\geq a, I\_{\mathbb{C}\_{i}}(y)\leq b, F\_{\mathbb{C}\_{i}}(y)\leq c\}.$$
  $\text{Therefore, for each } z\in U\_{\epsilon}$ 

*<sup>T</sup>*N*βx* (*z*) = D *TCi*(*x*)≥*a ICi*(*x*)≤*<sup>b</sup> FCi*(*x*)≤*c TCi*(*z*) ≥ D *TCi*(*y*)≥*a ICi*(*y*)≤*<sup>b</sup> FCi*(*y*)≤*c TCi*(*z*) = *T*N*βy* (*z*), *I*N*βx* (*z*) = E *TCi*(*x*)≥*a ICi*(*x*)≤*<sup>b</sup> FCi*(*x*)≤*c ICi*(*z*) ≤ E *TCi*(*y*)≥*a ICi*(*y*)≤*<sup>b</sup> FCi*(*y*)≤*c ICi*(*z*) = *I*N*βy* (*z*), *<sup>F</sup>*N*βx* (*z*) = E *TCi*(*x*)≥*a ICi*(*x*)≤*<sup>b</sup> FCi*(*x*)≤*c FCi*(*z*) ≤ E *TCi*(*y*)≥*a ICi*(*y*)≤*<sup>b</sup> FCi*(*y*)≤*c FCi*(*z*) = *F*N*βy* (*z*).

Hence, N *β y* ⊆ N *β x* . (⇐):For∈*U*,

$$\begin{aligned} \text{a. For any } \mathbf{x}, \mathbf{y} \in \mathcal{U}\_{\mathbf{y}} \text{ since } \widetilde{\mathbb{N}}\_{\mathbf{y}}^{\mathcal{S}} \subseteq \widetilde{\mathbb{N}}\_{\mathbf{x}}^{\mathcal{S}}\\ \text{or } \quad (\mathbf{x}) \sim \mathbf{x} \quad (\mathbf{x}) \sim \mathbf{x} \quad (\mathbf{x}) \sim \mathbf{x} \quad (\mathbf{x}) \sim \mathbf{x} \quad (\mathbf{x}) \sim \mathbf{x} \end{aligned}$$

*β*

$$T\_{\overline{\mathbb{N}}\_x^{\beta}}(y) \ge T\_{\overline{\mathbb{N}}\_y^{\beta}}(y) \ge a,\\ I\_{\overline{\mathbb{N}}\_x^{\beta}}(y) \le I\_{\overline{\mathbb{N}}\_y^{\beta}}(y) \le b \text{ and } F\_{\overline{\mathbb{N}}\_x^{\beta}}(y) \le F\_{\overline{\mathbb{N}}\_y^{\beta}}(y) \le c.$$

Therefore, N *β x* (*y*) ≥ *β*.

The notion of SVN *β*-neighborhood in the SVN *β*-covering approximation space in the following definition.

**Definition 5.** *Let* (*<sup>U</sup>*,*<sup>C</sup>* F ) *be a SVN β-covering approximation space and C* F = {*<sup>C</sup>*1, *C*2,..., *Cm*}*. For each x* ∈ *U, we define the β-neighborhood* N*βx of x as:*

$$\mathbb{N}\_x^\beta = \{ \mathcal{y} \in \mathcal{U} : \mathbb{N}\_x^\beta(\mathcal{y}) \ge \beta \}. \tag{3}$$

Note that N *β x* (*y*) is a SVN number *<sup>T</sup>*N*βx* (*y*), *I*N*βx*(*y*), *<sup>F</sup>*N*βx*(*y*) in Definition 5.

**Remark 2.** *Let C* F *be a SVN β-covering of U, β* = *a*, *b*, *c and C* F = {*<sup>C</sup>*1, *C*2,..., *Cm*}*. For each x* ∈ *U,*

$$\overline{\mathbb{N}}\_x^{\mathbb{R}} = \{ y \in \mathcal{U} : T\_{\overline{\mathbb{N}}\_x^{\beta}}(y) \ge a, I\_{\overline{\mathbb{N}}\_x^{\beta}}(y) \le b, F\_{\overline{\mathbb{N}}\_x^{\beta}}(y) \le c \}. \tag{4}$$

**Example 2** (Continued from Example 1)**.** *Let β* = 0.5, 0.3, 0.8*, then we have*

$$
\mathbb{N}\_{\mathbf{x}\_1}^{\mathcal{S}} = \{ \mathbf{x}\_1, \mathbf{x}\_2 \}, \\
\mathbb{N}\_{\mathbf{x}\_2}^{\mathcal{S}} = \{ \mathbf{x}\_2 \}, \\
\mathbb{N}\_{\mathbf{x}\_3}^{\mathcal{S}} = \{ \mathbf{x}\_3, \mathbf{x}\_5 \}, \\
\mathbb{N}\_{\mathbf{x}\_4}^{\mathcal{S}} = \{ \mathbf{x}\_2, \mathbf{x}\_4 \}, \\
\mathbb{N}\_{\mathbf{x}\_5}^{\mathcal{S}} = \{ \mathbf{x}\_5 \}.
$$

Some properties of the *β*-neighborhood in a SVN *β*-covering of *U* are presented in Theorem 2 and Proposition 2.

**Theorem 2.** *Let C* F *be a SVN β-covering of U and C* F = {*<sup>C</sup>*1, *C*2, ... , *Cm*}*. Then, the following statements hold:*

