# Decision-Making Approach Based on Neutrosophic Rough Information
## 2 Neutrosophic Rough Information
\begin{tabular}{c|c c c c c} \hline \(R\) & \(p\) & \(q\) & \(r\) & \(s\) & \(t\) \\ \hline \(p\) & _(1,1,0)_ & _(0.5,0.2,0.3)_ & _(0.1,0.9,0.4)_ & _(0.6,0.5,0.2)_ & _(0.2,0.1,0.8)_ \\ \(q\) & _(0.5,0.2,0.3)_ & _(1,1,0)_ & _(0.3,0.7,0.5)_ & _(0.1,0.9,0.6)_ & _(0.6,0.5,0.1)_ \\ \(r\) & _(0.1,0.9,0.4)_ & _(0.3,0.7,0.5)_ & _(1,1,0)_ & _(0.2,0.8,0.7)_ & _(0.1,0.9,0.6)_ \\ \(s\) & _(0.6,0.5,0.2)_ & _(0.1,0.9,0.6)_ & _(0.2,0.8,0.7)_ & _(1,1,0)_ & _(0.2,0.3,0.1)_ \\ \(t\) & _(0.2,0.1,0.8)_ & _(0.6,0.5,0.1)_ & _(0.1,0.9,0.6)_ & _(0.2,0.3,0.1)_ & _(1,1,0)_ \\ \hline \end{tabular}   
_Let \(X_{1}=\{(p,0.2,0.1,0.7),(q,0.4,0.5,0.6),(r,0.7,0.8,0.9),(s,0.2,0.9,0.1),(t,0.6,0.8, 0.4)\}\) be a neutrosophic set on \(X^{*}\). The lower and upper approximations of \(X_{1}\) are given as:_  
\[ \begin{array}{l}\underline{R}X_{1}=\{(p,0.2,0.1,0.7),(q,0.3,0.5,0.6),(r,0.4, 0.1,0.9),(s,0.2,0.5,0.6),(t,0.2,0.5,0.6)\},\\ \overline{R}X_{1}=\{(p,0.4,0.2,0.8),(q,0.6,0.9,0.4),(r,0.7,0.8,0.6),(s,0.2,0.9,0.1),(t,0.6,0.8,0.1)\}.\end{array} \]  
_Let \(Y^{*}=\{pr,qs,rt,sp,tq\}\subseteq X^{*}\times X^{*}\) and \(S\) a neutrosophic tolerance relation which is given as:_ \begin{tabular}{c|c c c c c} \hline \(S\) & \(pr\) & \(qs\) & \(rt\) & \(sp\) & \(tq\) \\ \hline \(pr\) & _(1,1,0)_ & _(0.2,0.2,0.7)_ & _(1,0.9,0.6)_ & _(0.1,0.5,0.4)_ & _(0.2,0.1,0.8)_ \\ \(qs\) & _(0.2,0.2,0.7)_ & _(1,1,0)_ & _(0.2,0.3,0.5)_ & _(0.1,0.5,0.6)_ & _(0.1,0.5,0.6)_ \\ \(rt\) & _(0.1,0.9,0.6)_ & _(0.2,0.3,0.5)_ & _(1,1,0)_ & _(0.2,0.1,0.8)_ & _(0.1,0.5,0.6)_ \\ \(sp\) & _(0.1,0.5,0.4)_ & _(0.1,0.5,0.6)_ & _(0.2,0.1,0.8)_ & _(1,1,0)_ & _(0.2,0.2,0.3)_ \\ \(tq\) & _(0.2,0.1,0.8)_ & _(0.1,0.5,0.6)_ & _(0.1,0.5,0.6)_ & _(0.2,0.2,0.3)_ & _(1,1,0)_ \\ \hline \end{tabular}   
_Let \(Y_{1}=\{(pr,0.2,0.1,0.5),(qs,0.1,0.3,0.3),(rt,0.2,0.1,0.4),(sp,0.1,0.1,0.2),( tq,0.1,0.4,0.3)\}\) be a neutrosophic set on \(Y^{*}\). The lower and upper approximations of \(Y_{1}\) are given as:_  
\[ \begin{array}{l}\underline{S}Y_{1}=\{(pr,0.2,0.1,0.5),(qs,0.1,0.3,0.3),(rt,0. 2,0.1,0.4),(sp,0.1,0.1,0.2),(tq,0.1,0.4,0.3)\},\\ \overline{S}Y_{1}=\{(pr,0.2,0.2,0.4),(qs,0.2,0.4,0.3),(rt,0.2,0.4,0.4),(sp,0.2, 0.3,0.2),(tq,0.2,0.4,0.3)\}.\end{array} \]  
_Thus, \(\underline{G}=(\underline{R}X_{1},\underline{S}Y_{1})\) and \(\overline{G}=(\overline{R}X_{1},\overline{S}Y_{1})\) are neutrosophic digraphs as shown in Figure 1._  
**Example 2**. : _Let \(X^{*}=\{u,v,w,x,y,z\}\) be a crisp set and \(R\) a neutrosophic tolerance relation on \(X^{*}\) given by_  
\begin{tabular}{c|c c c c c c} \hline \(R\) & \(u\) & \(v\) & \(w\) & \(x\) & \(y\) & \(z\) \\ \hline \(u\) & _(1,1,0)_ & _(0.2,0.3,0.5)_ & _(0.5,0.6,0.9)_ & _(0.3,0.8,0.3)_ & _(0.3,0.2,0.1)_ & _(0.1,0.1,0.5)_  
_Let \(X=\{(u,0.9,0.3,0.1),(v,0.5,0.6,0.2),(w,0.8,0.5,0.3),(x,0.7,0.6,0.9),(y,0.5,0.2,0.1),\)\((z,0.9,0.7,0.3)\}\) be a neutrosophic set on \(X^{*}\). Then the lower and upper approximations of \(X\) are given as follows:_  
\[\underline{RX}=\{(u,0.5,0.3,0.3),(v,0.5,0.2,0.3),(w,0.6,0.4,0.3),(x,0.7,0.3,0.9), (y,0.5,0.2,0.5),(z,0.5,0.5,0.8)\},\]  
\[\overline{RX}=\{(u,0.9,0.6,0.1),(v,0.8,0.7,0.1),(w,0.8,0.6,0.3),(x,0.8,0.7,0.3), (y,0.6,0.6,0.1),(z,0.9,0.7,0.2)\}.\]  
_Let \(Y^{*}=\{uv,vw,wx,xy,yz,zu,zw,vy\}\subseteq X^{*}\times X^{*}\) and \(S\) a neutrosophic tolerance relation on \(Y^{*}\) given as_