A Novel Rough Set Model in Generalized Single Valued Neutrosophic Approximation Spaces and Its Application

Abstract

In this paper, we extend the rough set model on two different universes in intuitionistic fuzzy approximation spaces to a single-valued neutrosophic environment. Firstly, based on the $(\alpha ,\beta ,\gamma )$-cut relation ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$, we propose a rough set model in generalized single-valued neutrosophic approximation spaces. Then, some properties of the new rough set model are discussed. Furthermore, we obtain two extended models of the new rough set model—the degree rough set model and the variable precision rough set model—and study some of their properties. Finally, we explore an example to illustrate the validity of the new rough set model.

1. Introduction

Smarandache [1,2] introduced the concept of the neutrosophic set (NS), which consists of three membership functions (truth membership function, indeterminacy membership function and falsity membership function), where each function value is a real standard or nonstandard subset of the nonstandard unit interval $[0^{-},1^{+}]$. The neutrosophic set generalizes the concepts of the classical set, fuzzy set [3], interval-valued fuzzy set [4], intuitionistic fuzzy set [5] and interval-valued intuitionistic fuzzy set [6]. The neutrosophic set model is an important tool for dealing with real scientific and engineering applications because it can handle not only incomplete information, but also the inconsistent information and indeterminate information that exist commonly in real situations.

For easily applying NSs in the real world, Smarandache [1] and Wang et al. [7] proposed single-valued neutrosophic sets (SVNSs) by simplifying NSs. SVNSs can also be seen as an extension of intuitionistic fuzzy sets [5], in which three membership functions are unrelated and their function values belong to the unit closed interval. SVNSs has been a hot research issue. Ye [8,9] proposed decision making based on correlation coefficients and weighted correlation coefficients of SVNSs and illustrated the application of the proposed methods. Baušys et al. [10] applied SVNSs to multi-criteria decision making and proposed a new extension of the crisp complex proportional assessment (COPRAS) method named COPRAS-SVNS. Zavadskas et al. [11] applied SVNSs to the weighted aggregated sum product assessment (WASPAS) method, named WASPAS-SVNS, and used the new method to solve sustainable assessment of alternative sites for the construction of a waste incineration plant. Zavadskas et al. [12] also applied WASPAS-SVNS to the selection of a lead-zinc flotation circuit design. Zavadskas et al. [13] proposed a single-valued neutrosophic multi-attribute market value assessment method and applied this method to the sustainable market valuation of Croydon University Hospital. Li et al. [14] applied the Heronian mean to the neutrosophic set, proposed some Heronian mean operators and illustrated their application in multiple attribute group decision making. Baušys and Juodagalvienė [15] demonstrated garage location selection for a residential house. In [16], Ye proposed similarity measures between interval neutrosophic sets and applied them to multi-criteria decision making problems under the interval neutrosophic environment. Ye [17] proposed three vector similarity measures of simplified neutrosophic sets and applied them to a multi-criteria decision making problem with simplified neutrosophic information. Majumdar and Samanta [18] studied the distance, similarity and entropy of SVNSs from a theoretical aspect. Peng et al. [19] developed a new outranking approach for multi-criteria decision making problems in the context of a simplified neutrosophic environment. Liu and Wang [20] introduced an interval neutrosophic prioritized ordered weighted aggregation operator w.r.t. interval neutrosophic numbers and discussed its application in multiple attribute decision making. To deal with difficulties in steam turbine fault diagnosis, Zhang et al. [21] investigated a single-valued neutrosophic multi-granulation rough set over two universes. Şahin [22] proposed two kinds of interval neutrosophic cross-entropies based on the extension of fuzzy cross-entropy and single-valued neutrosophic cross-entropy and developed two multi-criteria decision making methods using the interval neutrosophic cross-entropy. Ye [23] proposed similarity measures between SVNSs based on the tangent function and a multi-period medical diagnosis method based on the similarity measure and the weighted aggregation of multi-period information to solve multi-period medical diagnosis problems with single-valued neutrosophic information. Yang et al. [24] proposed SVNRs and studied some kinds of kernels and closures of SVNRs. Ye [25] presented a simplified neutrosophic harmonic averaging projection measure and its multiple attribute decision making method with simplified neutrosophic information. Stanujkic et al. [26] proposed a new extension of the multi-objective optimization (MULTIMOORA) method adapted for usage with a neutrosophic set.

Rough set theory, initiated by Pawlak [27,28], is a mathematical tool for the study of intelligent systems characterized by insufficient and incomplete information. The theory has been successfully applied to many fields, such as machine learning, knowledge acquisition, decision analysis, etc. To extend the application domain of rough set theory, more and more researchers have made some efforts toward the study of rough set models based on two different universes [29,30,31,32,33,34,35,36,37,38,39].

In recent years, many researchers have paid attention to combining neutrosophic sets with rough sets. Salama and Broumi [40] investigated the roughness of neutrosophic sets. Broumi and Smarandache put forward rough neutrosophic sets [41,42], as well as interval neutrosophic rough sets [43]. Yang et al. [44] proposed single-valued neutrosophic rough sets, which comprise a hybrid model of single-valued neutrosophic sets and rough sets. Along this line, this paper attempts to do some work regarding the fusion of single-valued neutrosophic sets and rough sets again. Concretely, we will extend the rough set model in [29] to a single-valued neutrosophic environment. Furthermore, we will apply the new model to a multi-attribute decision making problem.

The rest of this paper is organized as follows. In Section 2, we recall some basic notions related to Pawlak rough sets, SVNSs and single-valued neutrosophic rough sets. In Section 3, we propose a rough set model in generalized single-valued neutrosophic approximation spaces. Section 4 gives two extended models and studies some related properties. Section 5 explores an example to illustrate the new rough set model’s application in multi-attribute decision making. The last section summarizes the conclusions.

2. Preliminaries

In this section, we recall some basic notions of Pawlak rough sets, SVNSs and single-valued neutrosophic rough sets.

2.1. Pawlak Rough Sets

Definition 1.

([27,28]) Let U be a nonempty finite universe and R be an equivalence relation in U. $(U,R)$ is called a Pawlak approximation space. $\forall X\subseteq U$, the lower and upper approximations of X, denoted by $\underline{R}(X)$ and $\overline{R}(X)$, are defined as follows, respectively:

$$\begin{array}{c}\underline{R}(X)=\{x\in U\mid {[x]}_R\subseteq X\},\hfill \\ \overline{R}(X)=\{x\in U\mid {[x]}_R\cap X\ne \varnothing \},\hfill \end{array}$$

where ${[x]}_R=\{y\in U\mid (x,y)\in R\}$. $\underline{R}$ and $\overline{R}$ are called the lower and upper approximation operators, respectively. The pair $(\underline{R}(X),\overline{R}(X))$ is called a Pawlak rough set.

Furthermore, the positive region, boundary region and negative region of the subset X are defined by

$$\mathrm{pos}(X)=\underline{R}(X),\mathrm{neg}(X)=U-\overline{R}(X),\mathrm{bn}(X)=\overline{R}(X)-\underline{R}(X).$$

2.2. Single-Valued Neutrosophic Sets and Single-Valued Neutrosophic Rough Sets

Definition 2.

([7]) Let U be a space of points (objects), with a generic element in U denoted by x. A SVNS $\tilde{A}$ in U is characterized by three membership functions, a truth membership function $T_{\tilde{A}}$, an indeterminacy membership function $I_{\tilde{A}}$ and a falsity membership function $F_{\tilde{A}}$, where $\forall x\in U$, $T_{\tilde{A}}(x),I_{\tilde{A}}(x),F_{\tilde{A}}(x)\in [0,1]$.

The SVNS $\tilde{A}$ can be denoted by $\tilde{A}$ $=\{\langle x,T_{\tilde{A}}(x),I_{\tilde{A}}(x),F_{\tilde{A}}(x)\rangle \mid x\in U\}$ or $\tilde{A}$ $=(T_{\tilde{A}},I_{\tilde{A}},F_{\tilde{A}})$. $\forall x\in U$, $\tilde{A}$ $(x)=(T_{\tilde{A}}(x),I_{\tilde{A}}(x),F_{\tilde{A}}(x))$, and $(T_{\tilde{A}}(x),I_{\tilde{A}}(x),F_{\tilde{A}}(x))$ is called a single-valued neutrosophic number.

Definition 3.

([44]) An SVNS $\tilde{R}$ in $U\times U$ is called a single-valued neutrosophic relation (SVNR) in U, denoted by $\tilde{R}$ $=\{\langle (x,y),T_{\tilde{R}}(x,y),I_{\tilde{R}}(x,y),F_{\tilde{R}}(x,y)\rangle \mid (x,y)\in U\times U\}$, where $T_{\tilde{R}}:U\times U⟶[0,1]$, $I_{\tilde{R}}:U\times U⟶[0,1]$ and $F_{\tilde{R}}:U\times U⟶[0,1]$ denote the truth membership function, indeterminacy membership function and falsity membership function of $\tilde{R}$, respectively.

Definition 4.

([45]) Let $\tilde{R},\tilde{S}$ be two SVNRs in U. If $\forall x,y\in U$, $T_{\tilde{R}}(x,y)\le T_{\tilde{S}}(x,y)$, $I_{\tilde{R}}(x,y)\ge I_{\tilde{S}}(x,y)$ and $F_{\tilde{R}}(x,y)\ge F_{\tilde{S}}(x,y)$, then we say $\tilde{R}$ is contained in $\tilde{S}$, denoted by $\tilde{R}\tilde{\subseteq }\tilde{S}$. In other words, we say $\tilde{S}$ contains $\tilde{R}$, denoted by $\tilde{S}\tilde{\supseteq }\tilde{R}$.

Definition 5.

([24]) Let $\tilde{R}$ be an SVNR in U. If $\forall x\in U$, $T_{\tilde{R}}(x,x)=1$ and $I_{\tilde{R}}(x,x)=F_{\tilde{R}}(x,x)=0$, then $\tilde{R}$ is called a reflexive SVNR. If $\forall x,y\in U$, $T_{\tilde{R}}(x,y)=T_{\tilde{R}}(y,x)$, $I_{\tilde{R}}(x,y)=I_{\tilde{R}}(y,x)$ and $F_{\tilde{R}}(x,y)=F_{\tilde{R}}(y,x)$, then $\tilde{R}$ is called a symmetric SVNR. If $\forall x\in U$, ${\bigvee }_{y\in U}{T}_{\tilde{R}}(x,y)=1$ and ${\bigwedge }_{y\in U}{I}_{\tilde{R}}(x,y)={\bigwedge }_{y\in U}{F}_{\tilde{R}}(x,y)=0$, then $\tilde{R}$ is called a serial SVNR. If $\forall x,y,z\in U$, ${\bigvee }_{y\in U}(T_{\tilde{R}}(x,y)\wedge T_{\tilde{R}}(y,z))\le T_{\tilde{R}}(x,z)$, ${\bigwedge }_{y\in U}(I_{\tilde{R}}(x,y)\vee I_{\tilde{R}}(y,z))\ge I_{\tilde{R}}(x,z)$ and ${\bigwedge }_{y\in U}(F_{\tilde{R}}(x,y)\vee F_{\tilde{R}}(y,z))\ge F_{\tilde{R}}(x,z)$, then $\tilde{R}$ is called a transitive SVNR, where $“\vee ”$ and $“\wedge ”$ denote maximum and minimum, respectively.

Definition 6.

([24]) Let $\tilde{R}$ be an SVNR in U; the tuple $(U,\tilde{R})$ is called a single-valued neutrosophic approximation space. $\forall \tilde{A}\in$ SVNS$(U)$, the lower and upper approximations of $\tilde{A}$ w.r.t. $(U,\tilde{R})$, denoted by $\underline{\tilde{R}}(\tilde{A})$ and $\overline{\tilde{R}}(\tilde{A})$, are two SVNSs whose membership functions are defined as: $\forall x\in U$,

$T_{\underline{\tilde{R}}(\tilde{A})}(x)={\bigwedge }_{y\in U}(F_{\tilde{R}}(x,y)\vee T_{\tilde{A}}(y))$,

$I_{\underline{\tilde{R}}(\tilde{A})}(x)={\bigvee }_{y\in U}((1-I_{\tilde{R}}(x,y))\wedge I_{\tilde{A}}(y))$,

$F_{\underline{\tilde{R}}(\tilde{A})}(x)={\bigvee }_{y\in U}(T_{\tilde{R}}(x,y)\wedge F_{\tilde{A}}(y))$;

$T_{\overline{\tilde{R}}(\tilde{A})}(x)={\bigvee }_{y\in U}(T_{\tilde{R}}(x,y)\wedge T_{\tilde{A}}(y))$,

$I_{\overline{\tilde{R}}(\tilde{A})}(x)={\bigwedge }_{y\in U}(I_{\tilde{R}}(x,y)\vee I_{\tilde{A}}(y))$,

$F_{\overline{\tilde{R}}(\tilde{A})}(x)={\bigwedge }_{y\in U}(F_{\tilde{R}}(x,y)\vee F_{\tilde{A}}(y))$.

The pair $(\underline{\tilde{R}}(\tilde{A}),\overline{\tilde{R}}(\tilde{A}))$ is called the single-valued neutrosophic rough set of $\tilde{A}$ w.r.t. $(U,\tilde{R})$. $\underline{\tilde{R}}$ and $\overline{\tilde{R}}$ are referred to as the single-valued neutrosophic lower and upper approximation operators, respectively.

3. Rough Set Model in Generalized Single-Valued Neutrosophic Approximation Spaces

Guo et al. [29] studied the rough set model on two different universes in intuitionistic fuzzy approximation space. In this section, we will extend the rough set model in [29] to a single-valued neutrosophic environment.

Yang et al. [24] proposed the notions of single-valued neutrosophic relations from U to V and generalized single-valued neutrosophic approximation spaces as follows.

Definition 7.

([24]) Let U and V be two nonempty finite universes. The relation $\tilde{R}$ in $U\times V$ is called a single-valued neutrosophic relation from U to V, denoted by $\tilde{R}=\{\langle (x,y),T_{\tilde{R}}(x,y),I_{\tilde{R}}(x,y),F_{\tilde{R}}(x,y)\rangle |(x,y)\in U\times V\}$, where $T_{\tilde{R}}:U\times V⟶[0,1]$, $I_{\tilde{R}}:U\times V⟶[0,1]$ and $F_{\tilde{R}}:U\times V⟶[0,1]$ denote the truth membership function, indeterminacy membership function and falsity membership function of $\tilde{R}$, respectively.

The triple $(U,V,\tilde{R})$ is called a generalized single-valued neutrosophic approximation space on two different universes.

Remark 1.

If $U=V$, then we call $\tilde{R}$ a single-valued neutrosophic relation in U.

Definition 8.

Let $\tilde{R}$ be an SVNR from U to V. If $\forall x\in U$, $y\in V$, $T_{\tilde{R}}(x,y)=T_{\tilde{R}}(y,x)$, $I_{\tilde{R}}(x,y)=I_{\tilde{R}}(y,x)$ and $F_{\tilde{R}}(x,y)=F_{\tilde{R}}(y,x)$, then $\tilde{R}$ is called a symmetric SVNR. If $\forall x\in U$, ${\bigvee }_{y\in V}{T}_{\tilde{R}}(x,y)=1$ and ${\bigwedge }_{y\in V}{I}_{\tilde{R}}(x,y)={\bigwedge }_{y\in V}{F}_{\tilde{R}}(x,y)=0$, then $\tilde{R}$ is called a serial SVNR.

The union, intersection and containmentof two SVNRs from U to V are defined as follows, respectively.

Definition 9.

Let $R,S$ be two SVNRs from U to V.

(1) 

The union $R\tilde{\cup }S$ of R and S is defined by $R\tilde{\cup }S=\{\langle (x,y),$ max$\{T_R(x,y),T_S(x,y)\},$ min$\{I_R(x,y),I_S(x,y)\},$ min$\{F_R(x,y),F_S(x,y)\}\rangle |(x,y)\in U\times V\}$.

(2) 

The intersection $R\tilde{\cap }S$ of R and S is defined by $R\tilde{\cap }S=\{\langle (x,y),$ min$\{T_R(x,y),T_S(x,y)\},$ max$\{I_R(x,y),I_S(x,y)\},$ max$\{F_R(x,y),F_S(x,y)\}\rangle |(x,y)\in U\times V\}$.

(3) 

If $\forall (x,y)\in U\times V$, $T_R(x,y)\le T_S(x,y)$, $I_R(x,y)\ge I_S(x,y)$ and $F_R(x,y)\ge F_S(x,y)$, then we say R is contained in S, denoted by $R\tilde{\subseteq }S$.

Next, we give the notion of $(\alpha ,\beta ,\gamma )$-cut relation ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ of a single-valued neutrosophic relation $\tilde{R}$ from U to V.

Definition 10.

Let U, V be two nonempty finite universes and $\tilde{R}$ be a single-valued neutrosophic relation from U to V. For any $\alpha ,\beta ,\gamma \in (0,1]$, we define the $(\alpha ,\beta ,\gamma )$-cut relation ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ of $\tilde{R}$ as follows:

$${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}=\{(x,y)\in U\times V|T_{\tilde{R}}(x,y)\ge \alpha ,I_{\tilde{R}}(x,y)\le \beta ,F_{\tilde{R}}(x,y)\le \gamma \}.$$

According to Definition 10, if $(x,y)\in {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$, it indicates that the truth membership degree of the relationships of x and y w.r.t. SVNR $\tilde{R}$ is not less than $\alpha$, and the indeterminacy membership degree and falsity membership degree of the relationships of x and y w.r.t. SVNR $\tilde{R}$ are not more than $\beta$ and $\gamma$, respectively.

Definition 11.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is the $(\alpha ,\beta ,\gamma )$-cut relation defined in Definition 8. For any $x\in U,y\in V$, we define

$$\begin{array}{c}{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\{y\in V|T_{\tilde{R}}(x,y)\ge \alpha ,I_{\tilde{R}}(x,y)\le \beta ,F_{\tilde{R}}(x,y)\le \gamma \},\hfill \\ {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}(y)=\{x\in U|T_{\tilde{R}}(x,y)\ge \alpha ,I_{\tilde{R}}(x,y)\le \beta ,F_{\tilde{R}}(x,y)\le \gamma \}.\hfill \end{array}$$

The following Definition 12 gives a rough set model on two universes based on the $(\alpha ,\beta ,\gamma )$-cut relation ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ induced by a single-valued neutrosophic relation $\tilde{R}$ from U to V.

Definition 12.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. Suppose ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is the $(\alpha ,\beta ,\gamma )$-cut relation given in Definition 10 from U to V. For any set $Y\subseteq V$, the lower approximation and upper approximation of Y on two universes w.r.t. $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$ are defined by

$$\begin{array}{c}\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y and {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \},\hfill \\ \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing or {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}.\hfill \end{array}$$

The pair $(\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y),\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y))$ is called the rough set of Y w.r.t. $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$. If $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)=\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$, then Y is called the definable set w.r.t. $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$. If $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\ne \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$, then Y is called the undefinable set w.r.t. $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$.

Next, we define the positive region pos${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$, negative region neg${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$ and boundary region bn${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$ of Y, respectively:

$$\begin{array}{c}{{\mathit{pos}}_{\tilde{R}}}_{\{(\alpha ,\beta ,\gamma )\}}(Y)=\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y),\hfill \\ {{\mathit{neg}}_{\tilde{R}}}_{\{(\alpha ,\beta ,\gamma )\}}(Y)=U-\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y),\hfill \\ {{\mathit{bn}}_{\tilde{R}}}_{\{(\alpha ,\beta ,\gamma )\}}(Y)=\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)-\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y).\hfill \end{array}$$

Remark 2.

If $\tilde{R}$ is a series single-valued neutrosophic relation from U to V, i.e., $\forall x\in U,{\bigvee }_{y\in U}{T}_{\tilde{R}}(x,y)=1$ and ${\bigwedge }_{y\in U}{I}_{\tilde{R}}(x,y)={\bigwedge }_{y\in U}{F}_{\tilde{R}}(x,y)=0$, then there exists $y\in V$ such that $T_{\tilde{R}}(x,y)=1$ and $I_{\tilde{R}}(x,y)=0,F_{\tilde{R}}(x,y)=0$ for all $x\in U$ since V is finite. Therefore, for any $\alpha ,\beta ,\gamma \in (0,1]$, we have ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing$. Therefore, we have

$$\begin{array}{ccc}\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\hfill & =& \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y and {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \},\hfill \\ & =& \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\},\hfill \\ \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\hfill & =& \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \mathit{or} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \},\hfill \\ & =& \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \}.\hfill \end{array}$$

In the following, we discuss some properties of the lower approximation and the upper approximation given in Definition 12.

Theorem 1.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. Suppose ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is the $(\alpha ,\beta ,\gamma )$-cut relation given in Definition 10 from U to V. For any $Y,Y_1,Y_2\subseteq V$, the following properties hold:

(1) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y);$

(2) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(\varnothing )=\varnothing$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(V)=U$;

(3) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)=\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)=\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$;

(4) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)\supseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)\subseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$;

(5) 

If $Y_1\subseteq Y_2$, then $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$ and $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\subseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$;

(6) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)=\sim \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(\sim Y)$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)=\sim \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(\sim Y)$.

Proof. 

We only prove (3) and (6).

(3) $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y_1\cap Y_2$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y_1$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y_2$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y_1$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\cap \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y_2$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$

$=\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$;

 $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (Y_1\cup Y_2)\ne \varnothing$ or ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}$

$=\{x\in U|({\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y_1)\cup ({\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y_2)\ne \varnothing$ or ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y_1\ne \varnothing$ or ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}\cup \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y_2\ne \varnothing$ or ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}$

$=\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$.

(6) $\sim \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(\sim Y)$

$=\sim \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (\sim Y)\ne \varnothing$ or ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (\sim Y)=\varnothing$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$

$=\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$;

 $\sim \underline{{\tilde{R}}_{\{}(\alpha ,\beta ,\gamma )\}}(\sim Y)$

$=\sim \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq (\sim Y)$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$

$=\sim \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y=\varnothing$ and $x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$

$=\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing$ or ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}$

$=\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )}}(Y)$. ☐

Remark 3.

In general,

(1) 

$\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(\varnothing )\ne \varnothing$, $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(V)\ne U$;

(2) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)\ne \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$ and $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)\ne \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$,

as shown in the following example.

Example 1.

Let $U=\{x_1,x_2,x_3\}$, $V=\{y_1,y_2,y_3,y_4\}$. $Y_1=\{y_1\}$ and $Y_2=\{y_3\}$. The single-valued neutrosophic relation $\tilde{R}$ from U to V is given in Table 1.

(1) 

Take $\alpha =0.9$, $\beta =0.3$ and $\gamma =0.2$; we have ${\tilde{R}}_{\{(0.9,0.3,0.2)\}}(x_1)=\{y_4\}$, ${\tilde{R}}_{\{(0.9,0.3,0.2)\}}(x_2)=\{y_1\}$, ${\tilde{R}}_{\{(0.9,0.3,0.2)\}}(x_3)=\{y_3\}$.

By Definition 12, we have $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(\varnothing )=\{x_3\}\ne \varnothing$ and $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(V)=\{x_1,x_2\}\ne U$.

(2) 

Take $\alpha =0.5$, $\beta =0.3$ and $\gamma =0.2$; we have ${\tilde{R}}_{\{(0.5,0.3,0.2)\}}(x_1)=\{y_1,y_2,y_3,y_4\}$, ${\tilde{R}}_{\{(0.5,0.3,0.2)\}}(x_2)=\{y_1,y_3\}$, ${\tilde{R}}_{\{(0.5,0.3,0.2)\}}(x_3)=\varnothing$.

By Definition 12, we have

$$\begin{array}{c}ll\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)=\varnothing ,\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)=\{x_3\},\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)=\{x_2,x_3\}.\hfill \\ \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)=\{x_1,x_2\},\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)=\{x_1,x_2,x_3\},\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)=\varnothing .\hfill \end{array}$$

Obviously, $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)\ne \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$ and $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)\ne \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)$.

Theorem 2.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. ${\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}$ and ${\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}$ are two relations defined in Definition 10. If $\tilde{R}$ is a series, ${\alpha }_1\le {\alpha }_2$, ${\beta }_1\ge {\beta }_2$ and ${\gamma }_1\ge {\gamma }_2$, then

$(1)$ $\underline{{\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}}(Y)$;

$(2)$ $\overline{{\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}}(Y)\subseteq \overline{{\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}}(Y)$.

Proof. 

$(1)$ Since ${\alpha }_1\le {\alpha }_2$, ${\beta }_1\ge {\beta }_2$ and ${\gamma }_1\ge {\gamma }_2$, for any $x\in U$, we have

$$\begin{array}{c}{\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}(x)=\{y\in V|T_{\tilde{R}}(x,y)\ge {\alpha }_1,I_{\tilde{R}}(x,y)\le {\beta }_1,F_{\tilde{R}}(x,y)\le {\gamma }_1\}\hfill \\ \supseteq \{y\in V|T_{\tilde{R}}(x,y)\ge {\alpha }_2,I_{\tilde{R}}(x,y)\le {\beta }_2,F_{\tilde{R}}(x,y)\le {\gamma }_2\}\hfill \\ ={\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}(x)\hfill \end{array}$$

By Definition 12, for any $x\in \underline{{\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}}(Y)$, we have ${\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}(x)\subseteq Y$. Thus ${\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}(x)\subseteq {\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}(x)\subseteq Y$, which implies that $x\in \underline{{\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}}(Y).$ Hence, $\underline{{\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}}(Y)$.

$(2)$ By (1), for any $x\in U$, we have ${\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}(x)\supseteq {\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}(x)$.

So

$${\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}(x)\cap Y\supseteq {\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}(x)\cap Y for any x\in U.$$

By Definition 12, for any $x\in \overline{{\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}}(Y)$, we have ${\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}(x)\cap Y\ne \varnothing$. Thus, ${\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}(x)\cap Y\ne \varnothing$, which implies that $x\in \overline{{\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}}(Y)$. Hence, $\overline{{\tilde{R}}_{\{({\alpha }_1,{\beta }_1,{\gamma }_1)\}}}(Y)\subseteq \overline{{\tilde{R}}_{\{({\alpha }_2,{\beta }_2,{\gamma }_2)\}}}(Y)$. ☐

Theorem 3.

Let $\tilde{R},\tilde{S}$ be two series single-valued neutrosophic relations from U to V. If $\tilde{R}\tilde{\subseteq }\tilde{S}$, then $\forall Y\subseteq V$, we have:

(1) 

$\underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$;

(2) 

$\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \overline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$.

Proof. 

(1) Since $\tilde{R}\tilde{\subseteq }\tilde{S}$, we have

$$\begin{array}{c}{T}_{\tilde{S}}(x,y)\ge T_{\tilde{R}}(x,y),I_{\tilde{S}}(x,y)\le I_{\tilde{R}}(x,y) and F_{\tilde{S}}(x,y)\le F_{\tilde{R}}(x,y) for any (x,y)\in U\times V.\hfill \\ Then,{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\{y\in V|T_{\tilde{S}}(x,y)\ge \alpha ,I_{\tilde{S}}(x,y)\le \beta ,F_{\tilde{S}}(x,y)\le \gamma \}\hfill \\ \supseteq \{y\in V|T_{\tilde{R}}(x,y)\ge \alpha ,I_{\tilde{R}}(x,y)\le \beta ,F_{\tilde{R}}(x,y)\le \gamma \}\hfill \\ ={\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x).\hfill \end{array}$$

By Definition 12, for any $x\in \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$, we have ${\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y$. Thus, ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y$, which implies that $x\in \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y).$ Hence, $\underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$.

$(2)$ By $(1)$, for any $x\in U$, we have ${\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\supseteq {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)$. Thus, ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\subseteq {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y$ for any $x\in U$. By Definition 12, for any $x\in \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$, we have ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing$. Thus, ${\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing$, which implies that $x\in \overline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$. Hence, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \overline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$. ☐

Lemma 1.

Let $\tilde{R},\tilde{S}$ be two single-valued neutrosophic relations from U to V. For any $x\in U$ and $\alpha ,\beta ,\gamma \in (0,1]$, we have:

(1)

${(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)\supseteq {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cup {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)$;

(2)

${(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)={\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)$.

Proof. 

(1)

For any $x\in U$, we have:

$\begin{array}{l}{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\{y\in V|\max (T_{\tilde{R}}(x,y),T_{\tilde{S}}(x,y))\ge \alpha ,\min (I_{\tilde{R}}(x,y),I_{\tilde{S}}(x,y))\le \beta ,\\ \min (F_{\tilde{R}}(x,y),F_{\tilde{S}}(x,y))\le \gamma \}\\ \supseteq \{y\in V|T_{\tilde{R}}(x,y)\ge \alpha ,I_{\tilde{R}}(x,y)\le \beta ,F_{\tilde{R}}(x,y)\le \gamma \}\\ \cup \{y\in V|T_{\tilde{S}}(x,y)\ge \alpha ,I_{\tilde{S}}(x,y)\le \beta ,F_{\tilde{S}}(x,y)\le \gamma \}\\ ={\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cup {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\end{array}$

(2)

For any $x\in U$, we have:

$\begin{array}{c}\begin{array}{cc}{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)\hfill & =\{y\in V|\min (T_{\tilde{R}}(x,y),T_{\tilde{S}}(x,y))\ge \alpha ,\max (I_{\tilde{R}}(x,y),I_{\tilde{S}}(x,y))\le \beta ,\hfill \\ & \max (F_{\tilde{R}}(x,y),F_{\tilde{S}}(x,y))\le \gamma \}\hfill \\ & =\{y\in V|T_{\tilde{R}}(x,y)\ge \alpha ,I_{\tilde{R}}(x,y)\le \beta ,F_{\tilde{R}}(x,y)\le \gamma \}\hfill \end{array}\hfill \\ \begin{array}{cc}\cap \{y\in V|T_{\tilde{S}}(x,y)\ge \alpha ,I_{\tilde{S}}(x,y)\le \beta ,F_{\tilde{S}}(x,y)\le \gamma \}\hfill & \\ ={\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\hfill \end{array}\hfill \end{array}$

☐

Theorem 4.

Let $\tilde{R},\tilde{S}$ be two series single-valued neutrosophic relations from U to V. For any $Y\subseteq V$ and $\alpha ,\beta ,\gamma \in (0,1]$, we have:

(1) 

$\underline{{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cap \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cup \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$;

(2) 

$\overline{{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\supseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cup \overline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$.

Proof. 

(1)

By Lemma 1 (1), we have

$\begin{array}{cc}\underline{{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\hfill & =\{x\in U|{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\}\hfill \\ & \subseteq \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cup {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\}\hfill \\ & =\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\}\cap \{x\in U|{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\}\hfill \\ & =\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cap \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y).\hfill \end{array}$

So

$\underline{{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cap \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cup \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$.

(2)

By Lemma 1 (1), we have

$\begin{array}{cc}\overline{{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\hfill & =\{x\in U|{(\tilde{R}\tilde{\cup }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \}\hfill \\ & \supseteq \{x\in U|({\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cup {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x))\cap Y\ne \varnothing \}\hfill \\ & =\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \}\cup \{x\in U|{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \}\hfill \\ & =\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cup \overline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y).\hfill \end{array}$

☐

Theorem 5.

Let $\tilde{R},\tilde{S}$ be two series single-valued neutrosophic relations from U to V. For any $Y\subseteq V$ and $\alpha ,\beta ,\gamma \in (0,1]$, we have:

(1) 

$\underline{{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\supseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cup \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\supseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cap \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$;

(2) 

$\overline{{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\subseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cap \overline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$.

Proof. 

(1)

By Lemma 1 (2), we have

$\begin{array}{cc}\underline{{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\hfill & =\{x\in U|{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\}\hfill \\ & =\{x\in U|({\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x))\subseteq Y\}\hfill \\ & \supseteq \{x\in U|({\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\}\cup \{x\in U|({\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y\}\hfill \\ & =\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cup \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y).\hfill \end{array}$

Therefore,

$\underline{{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\supseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cup \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\supseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cap \underline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y).$

(2)

By Lemma 1 (2), we have

$\begin{array}{cc}\overline{{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\hfill & =\{x\in U|{(\tilde{R}\tilde{\cap }\tilde{S})}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \}\hfill \\ & =\{x\in U|({\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}\cap {\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}})(x)\cap Y\ne \varnothing \}\hfill \\ & \subseteq \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \}\cap \{x\in U|{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \}\hfill \\ & =\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\cap \overline{{\tilde{S}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y).\hfill \end{array}$

☐

Next, we define the inverse lower approximation and upper approximation on two universes w.r.t. $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$ as follows:

Definition 13.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. For any $\alpha ,\beta ,\gamma \in (0,1],$ $X\subseteq U$, the inverse lower approximation and upper approximation of X on two universes w.r.t. $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$ are defined as:

$$\begin{array}{cc}\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X)\hfill & =\{y\in V|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}(y)\subseteq X and {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}(y)\ne \varnothing \},\hfill \\ \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X)\hfill & =\{y\in V|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}(y)\cap X\ne \varnothing or {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}(y)=\varnothing \}.\hfill \end{array}$$

The pair $(\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X),\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X))$ is called the inverse rough set of X w.r.t. $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$.

Theorem 6.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is the $(\alpha ,\beta ,\gamma )$-cut relation given in Definition 10 from U to V, where $\alpha ,\beta ,\gamma \in (0,1]$. For any $X,X_1,X_2\subseteq U$, we have:

(1) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X)\subseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X);$

(2) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(\varnothing )=\varnothing$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(U)=V$;

(3) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1\cap X_2)=\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_2)$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1\cup X_2)=\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_2)$;

(4) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1\cup X_2)\supseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_2)$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1\cap X_2)\subseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_2)$;

(5) 

If $X_1\subseteq X_2$, then $\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1)\subseteq \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_2)$ and $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_1)\subseteq \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X_2)$;

(6) 

$\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X)=\sim \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(\sim X)$, $\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(X)=\sim \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{-1}}(\sim X)$.

Proof. 

The proof is similar to that of Theorem 1.  ☐

Definition 14.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is a $(\alpha ,\beta ,\gamma )$-cut relation defined in Definition 10. For any $Y\subseteq V$, the approximate precision ${\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$ of Y w.r.t. ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is defined as follows:

$${\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)=\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)|}{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)|},$$

where $|Y|$ represents the cardinality of the set Y.

Let ${\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)=1-{\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$, and ${\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)$ is called the rough degree of with regard to ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$. It is obviously that $0\le {\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\le 1$ and $0\le {\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y)\le 1$.

The following Theorem 7 discusses the properties of approximation precision and rough degree.

Theorem 7.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space. ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is a $(\alpha ,\beta ,\gamma )$-cut relation defined in Definition 10. For any $Y_1,Y_2\subseteq V(Y_1\ne \varnothing ,Y_2\ne \varnothing )$, then the rough degree and the approximate precision of the set $Y_1,Y_2,Y_1\cup Y_2,$ and $Y_1\cap Y_2$ satisfy the following properties:

(1) 

$$\begin{gathered} {\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\le {\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|+ \\ {\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|; \end{gathered}$$

(2) 

$$\begin{gathered} {\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\ge {\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|+ \\ {\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-{\rho }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(A\cap Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|. \end{gathered}$$

Proof. 

According to the definition of the rough degree, we have

$$\begin{array}{cc}{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)\hfill & =1-\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|}{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|}\hfill \\ & =1-\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|}{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|}\hfill \\ & \le 1-\frac{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|}{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|}.\hfill \end{array}$$

Then, we have

$$\begin{array}{c}{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\hfill \\ \le |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)\hfill & =1-\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)|}{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)|}\hfill \\ & =1-\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|}{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)|}\hfill \\ & =1-\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)}{|\stackrel{̲}{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \stackrel{̲}{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|}.\hfill \end{array}$$

Hence,

${\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\le |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|.$

Furthermore, we know $|A\cup B|=|A|+|B|-|A\cap B|$ holds for any sets A and B. Then,

$$\begin{array}{c}{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\hfill \\ \le |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cup \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\hfill \\ =|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|+|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\stackrel{̲}{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \stackrel{̲}{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|\hfill \\ -|\underset{¯}{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|+|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\hfill \\ =|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|+|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-(|\stackrel{̲}{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\hfill \\ \cap |\underset{¯}{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|)\hfill \\ \le |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|+|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)\hfill \\ |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|.\hfill \end{array}$$

Furthermore, by

$$\begin{gathered} \begin{array}{c}{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)=1-\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|}{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|}and{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)=1-\frac{|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|}{|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|},wehave\hfill \\ |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|={\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|and\hfill \end{array} \\ |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|={\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|. \end{gathered}$$

Therefore,

$$\begin{array}{c}{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cup Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\hfill \\ \le |\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|+|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|-|\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\hfill \\ -{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|\hfill \\ ={\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)|+{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|\hfill \\ -{\mu }_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1\cap Y_2)|\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_1)\cap \overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y_2)|.\hfill \end{array}$$

 ☐

4. Two Extended Models

In this section, we give two extended rough set models of the model in Section 3, i.e., the degree rough set model and the variable precision rough set model.

4.1. The Degree Rough Set Model on Two Different Universes

Definition 15.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ be a $(\alpha ,\beta ,\gamma )$-cut relation defined in Definition 10. For any $Y\subseteq V$, we define the degree lower and upper approximations of Y w.r.t. the degree k, $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$ as follows:

$\begin{array}{cc}{\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)\hfill & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|\le k \mathit{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)|-|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y|\le k \mathit{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \};\hfill \\ {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)\hfill & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y|>k \mathit{or} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}.\hfill \end{array}$

where k is a finite nonnegative integer and $|Y|$ denotes the cardinality of the set Y.

The pair $({\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y),{\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y))$ is called the degree rough set of Y w.r.t. the degree k, $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$.

We also define the positive region pos${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^k}(Y)$, negative region neg${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^k}(Y)$ and boundary region bn${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^k(Y)}$ of Y as follows:

pos${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^k}(Y)={\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)$,

neg${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^k}(Y)=U-{\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)$,

bn${}_{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^k}(Y)={\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)-{\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)$.

Remark 4.

In Definition 15, if $k=0$, then

$$\begin{array}{cc}{\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y)\hfill & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|\le 0 \mathit{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|=0 \mathit{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & =\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\subseteq Y \mathit{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & =\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y);\hfill \\ {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y)\hfill & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y|>0 \mathit{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}\hfill \\ & =\{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y\ne \varnothing \mathit{or} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}\hfill \\ & =\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}(Y),\hfill \end{array}$$

which implies that the lower and upper approximation operators in Definition 12 are special cases of the degree lower and upper approximation operators in Definition 15, respectively.

In following Theorem 8, we discuss some properties of the degree lower and upper approximation operators.

Theorem 8.

Let $(U,V,\tilde{R})$ be a generalized single-valued neutrosophic approximation space, and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}$ is the $(\alpha ,\beta ,\gamma )$-cut relation defined in Definition 10. For any $Y,Y_1,Y_2\subseteq V,$ we have:

(1) 

${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y_1\cap Y_2)={\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y_1)\cap {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y_2)$, ${\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y_1\cup Y_2)={\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y_1\cup {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y_2$;

(2) 

${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1\cap Y_2)\subseteq {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1)\cap {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$, ${\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1\cup Y_2)\supseteq {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1)\cup {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$;

(3) 

${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1\cup A_2)\supseteq {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^0(Y_1)\cup {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$, ${\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1\cap Y_2)\subseteq {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1)\cap {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$;

(4) 

${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)=\sim {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(\sim Y)$, ${\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)=\sim {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(\sim Y)$;

(5) 

If $Y_1\subseteq Y_2$, then ${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1)\subseteq {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$ and ${\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1)\subseteq {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$;

(6) 

If $k\ge l$, then ${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^l(Y)\subseteq {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)$ and ${\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)\subseteq {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^l(Y)$, where k is a finite positive integer.

Proof. 

We only prove $(4)$ and $(6)$.

(4) Notice that $Y_1-Y_2=Y_1\cap (\sim Y_2)$ and $\sim (\sim Y)=Y$ for any set $Y\in V$; we have

$\begin{array}{cc}\sim {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(\sim Y)\hfill & =\sim \{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (\sim Y)|>k \mathrm{or} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}\hfill \\ & =\sim \{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|>k \mathrm{or} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}\hfill \\ & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (\sim Y)|\le k \mathrm{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & ={\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y),\hfill \\ \sim {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(\sim Y)\hfill & =\sim \{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-(\sim Y)|\le k \mathrm{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & =\sim \{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (Y)|\le k \mathrm{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (Y)|>k \mathrm{or} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}\hfill \\ & ={\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y).\hfill \end{array}$

(6) Since $k\ge l$, for any $x\in U$ and $Y\subseteq V$, we have

$\begin{array}{cc}{\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^l(Y)\hfill & =\{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|\le l \mathrm{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & \subseteq \{x\in U||{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|\le k \mathrm{and} {\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\hfill \\ & ={\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y).\hfill \end{array}$

For any $x\in {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^l(Y)$, we have $|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|\le l$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing$, then we have $|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)-Y|\le k$ and ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing$, so $x\in {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)$. Hence, ${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^l(Y)\subseteq {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y)$.  ☐

Remark 5.

In general, ${\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1\cap Y_2)\supseteq {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1)\cap {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$ does not hold, where k is a finite positive integer. The reason is that if $k>0$, then $|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}-Y_1|\le k$ and $|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}-Y_2|\le k\neg ⟹|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}-Y_1\cap Y_2|\le k.$ Besides, ${\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k$ $(Y_1\cup Y_2)$ $\subseteq {\overline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_1)\cup {\underline{{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^k(Y_2)$ also does not hold in general, where k is a finite positive integer. The reason is that if $k>0$, then $|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap (Y_1\cup Y_2)|>k\neg ⟹|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}\cap Y_1|>k$ or $|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}\cap Y_2|>k$.

4.2. The Variable Precision Rough Set Model on Two Different Universes

Definition 16.

Let $(U,V,\tilde{R})$ be a generalized single valued neutrosophic approximation space. For any $Y\subseteq V,\alpha ,\beta ,\gamma \in (0,1]$, we define the variable precision lower and upper approximation of Y w.r.t. the control parameter ν, $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$ as follows, respectively:

${\underline{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^{\nu }(Y)=\{x\in U|\frac{|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y|}{|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)|}\ge 1-\nu ,{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}$,

${\overline{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^{\nu }(Y)=\{x\in U|\frac{|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\cap Y|}{|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)|}>\nu ,{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)\ne \varnothing \}\cup \{x\in U|{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}(x)=\varnothing \}$,

where $\nu \in [0,0.5)$, $|Y|$ denotes the cardinality of the set Y.

The pair $({\underline{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^{\nu },{\overline{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^{\nu })$ is called the variable precision single-valued neutrosophic rough set of A with regard to the control parameter $\nu$, $(U,V,\tilde{R})$ and $(\alpha ,\beta ,\gamma )$.

We also define the positive region pos${}_{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{\nu }}(Y)$, negative region neg${}_{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{\nu }}(Y)$ and boundary region bn${}_{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{\nu }}(Y)$ of Y about ${\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{\nu }(Y)$ as follows:

pos${}_{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{\nu }}(Y)={\underline{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^{\nu }(Y)$,

neg${}_{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{\nu }}(Y)=U-{\overline{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^{\nu }(Y)$,

bn${}_{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}^{\nu }}(Y)={\overline{V{\tilde{R}}_{\{(\alpha ,\beta ,\gamma )\}}}}^{\nu }(Y)-{\underset{}{V{\stackrel{}{R}}_{}}}^{}$