## 3 Methodology
### Pythagorean Fuzzy Rough Approximation Construction
Definition 6: Let \(PFDS=(U,AT=C\cup D,V,f)\) be a Pythagorean fuzzy decision system, where \(C\) and \(D\) are the condition attribute set and decision attribute set, respectively. The condition attribute set \(B\subseteq C\), \(x=\left\{f(x,c)|c\in B\right\}\) and \(y=\left\{f(y,c)|c\in B\right\}\) are PFSs of the object \(x,y\in U\) under the condition attribute set \(B\), where \(f(x,c)=\left\{c,\mu_{x}(c),\
u_{x}(c)\right\}\) and \(f(y,c)=\left\{c,\mu_{y}(c),\
u_{y}(c)\right\}\) are PFNs of the objects \(x\) and \(y\), respectively, under the condition attribute \(c\). The Pythagorean fuzzy closeness measure \(\vartheta(x,y)\) between the objects \(x\) and \(y\) under the condition attribute set \(B\) in the Pythagorean fuzzy decision system is defined as  
\[\vartheta(x,y)=\left(\sqrt{S_{\vartheta}(x,y)},\sqrt{D_{\vartheta}(x,y)}\right) \tag{5}\]  
where \(S_{\vartheta}(x,y)\) and \(D_{\vartheta}(x,y)\) are the similarity measure and divergence measure between the two PFSs \(x\) and \(y\), respectively. It is easy to verify Definition 6 has the following properties:  
Property 2: Let \(PFDS=(U,AT=C\cup D,V,f)\) be a Pythagorean fuzzy decision system and \(\vartheta(x,y)\) be the closeness measure between the objects \(x\) and \(y\) under the attribute set \(B\subseteq C\) in the Pythagorean fuzzy decision system. For \(\forall c\in B\), then  
(1) \(f(x,c)=f(y,c)\), if and only if \(\vartheta(x,y)=(1,0)\);  
(2) if \(f(x,c)=(1,0)\) and \(f(y,c)=(0,1)\), then \(\vartheta(x,y)=(0,1)\);_(3) if \(f(x,c)=(1,0)\) and \(f(y,c)=\left(\sqrt{2}/2,\sqrt{2}/2\right)\), then \(\vartheta(x,y)=\left(\sqrt{2}/2,\sqrt{2}/2\right)\); (4) if \(f(x,c)=\left(\sqrt{2}/2,\sqrt{2}/2\right)\) and \(f(y,c)=(0,1)\), then \(\vartheta(x,y)=\left(\sqrt{2}/2,\sqrt{2}/2\right)\)._  
Obviously, the conclusion in Property 2 is consistent with human intuition. For example, (1) can be interpreted as the sum of PFSs \(x\) and \(y\) is equal if and only if the similarity measure between \(x\) and \(y\) is 1 and the divergence measure is 0. Therefore, the concept of the \((\alpha,\beta)\)-level cut set under the Pythagorean fuzzy closeness measure is given below. **Definition 7**. : _Let \(PFDS=(U,AT=C\cup D,V,f)\) be the Pythagorean fuzzy decision system and \(\vartheta(x,y)=(S_{\vartheta}(x,y),D_{\vartheta}(x,y))\) be the Pythagorean fuzzy closeness measure of the objects \(x\) and \(y\) related to the condition attribute set \(B\subseteq C\). For any \(\alpha,\beta\in[0,1]\), \(\alpha\) and \(\beta\) satisfy \(0\leq\alpha+\beta\leq 1\), and the \((\alpha,\beta)\)-level cut set \(\vartheta^{\beta}_{\alpha}\) under Pythagorean fuzzy closeness measure \(\vartheta\) is defined as_  
\[\vartheta^{\beta}_{\alpha}=\left\{(x,y)\in U\times U\middle|\sqrt{S_{\vartheta }(x,y)}\geq\alpha\text{ and}\sqrt{D_{\vartheta}(x,y)}\leq\beta\right\} \tag{6}\]  
\[\vartheta^{\beta}_{\alpha}=\left\{y\in U\middle|\left(x,y\right)\in\vartheta^ {\beta}_{\alpha}\right\} \tag{7}\]  
_Otherwise, \(\vartheta_{\alpha}=\left\{(x,y)\in U\times U\middle|\sqrt{S_{\vartheta}(x,y)} \geq\alpha\right\}\) and \(\vartheta^{\beta}=\left\{(x,y)\in U\times U\middle|\sqrt{D_{\vartheta}(x,y)} \right.\)\(\leq\beta\}\) are the \(\alpha\)-level cut set similar to \(\vartheta\) and the \(\beta\)-level cut set diverges to \(\vartheta\). Obviously, \(\vartheta^{\beta}_{\alpha}\) is a classical binary relation on the universe \(U\)._  
From Definition 7, \(\alpha\) and \(\beta\) represent the minimum and maximum thresholds of similarity measures between the objects \(x\) and \(y\) related to condition attribute set \(B\), respectively. In the actual decision, the threshold values of \(\alpha\) and \(\beta\) are determined by experts according to the actual needs of similarity and divergence. Then the \((\alpha,\beta)\)-level cut set \(\vartheta^{\beta}_{\alpha}\) has the following property. **Property 3**.