## 3 Methodology
### Pythagorean Fuzzy Rough Approximation Construction
**_Let \(A\) and \(B\) be two PFSs on the universe \(U\), and then the similarity measure \(S(A,B)\) between the two PFSs \(A\) and \(B\) is defined as_  
\[S(A,B)=\min_{x\in U}\biggl{\{}1-\frac{1}{2}\Bigl{(}\left|\mu_{A}^{2}(x)-\mu_{ B}^{2}(x)\right|\ +\left|\
u_{A}^{2}(x)-\
u_{B}^{2}(x)\right|+\left|\pi_{A}^{2}(x)-\pi_{B}^{2}(x) \right|\Bigr{)}\biggr{\}} \tag{2}\]  
_where \(\pi_{A}(x)\) and \(\pi_{B}(x)\) are the hesitance degrees of the two PFSs \(A\) and \(B\)._  
The proof of properties of Theorem 1 is shown in Appendix A. It can be seen from Theorem 1 that the similarity measure of the two PFSs is given based on Definition 5, which is the minimum of the similarity of the corresponding PFNs in the two PFSs. Definition 5 is consistent with the smaller membership value in the intersection operation of PFSs. **Theorem 2. **_Let \(A\) and \(B\) be two PFSs on the universe \(U\), and then the divergence measure of the two PFSs \(A\) and \(B\) is defined as_  
\[D(A,B)=\max_{x\in U}\biggl{\{}\frac{1}{2}\Bigl{(}\left|\mu_{A}^{2}(x)-\mu_{B} ^{2}(x)\right|+\left|\
u_{A}^{2}(x)-\
u_{B}^{2}(x)\right|\Bigr{)}\biggr{\}} \tag{3}\]  
The proof of properties of Theorem 2 is shown in Appendix A. Similar to the explanation of Theorem 1, the new definition of the divergence measure between the two PFSs is given as follows. **Theorem 3. ** Let \(S(A,B)\) and \(D(A,B)\) be the similarity measure and the divergence measure, respectively, between the two PFSs \(A\) and \(B\), then the closeness measure between the two PFSs \(A\) and \(B\) is defined as  
\[\vartheta(A,B)=\left(\sqrt{S(A,B)},\sqrt{D(A,B)}\right) \tag{4}\]  
**Proof. ** It should satisfy the four properties in Definition 4. The proofs are as follows. (1) From Property 1, \(0\leq S(A,B)\leq 1\) and \(0\leq D(A,B)\leq 1\). Otherwise,  
\[ \begin{array}{l}S(A,B)+D(A,B)\\ =\min\limits_{x\in U}\left\{1-\frac{1}{2}\left(\left|\mu_{A}^{2}(x)-\mu_{B}^{2} (x)\right|+\left|\
u_{A}^{2}(x)-\
u_{B}^{2}(x)\right|+\left|\pi_{A}^{2}(x)-\pi _{B}^{2}(x)\right|\right)\right\}\\ +\max\limits_{x\in U}\left\{\frac{1}{2}\left(\left|\mu_{A}^{2}(x)-\mu_{B}^{2} (x)\right|+\left|\
u_{A}^{2}(x)-\
u_{B}^{2}(x)\right|\right)\right\}\\ =1-\max\limits_{x\in U}\left\{\frac{1}{2}\left(\left|\mu_{A}^{2}(x)-\mu_{B}^{2 }(x)\right|+\left|\
u_{A}^{2}(x)-\
u_{B}^{2}(x)\right|+\left|\pi_{A}^{2}(x)- \pi_{B}^{2}(x)\right|\right)\right\}\\ +\max\limits_{x\in U}\left\{\frac{1}{2}\left(\left|\mu_{A}^{2}(x)-\mu_{B}^{2 }(x)\right|+\left|\
u_{A}^{2}(x)-\
u_{B}^{2}(x)\right|\right)\right\}\\ \leq 1\end{array} \]  
Therefore, \(\vartheta(A,B)\) is a PFN. (2) \(\vartheta(A,B)=(1,0)\), if and only if \(S(A,B)=1\) and \(D(A,B)=0\), if and only if \(A=B\);  
(3) Obviously, \(\vartheta(A,B)=\vartheta(B,A)\);  
(4) if \(A\subseteq B\subseteq C\), then for \(\forall x\in U,\mu_{A}(x)\leq\mu_{B}(x)\leq\mu_{C}(x)\) and \(\
u_{A}(x)\geq\
u_{B}(x)\geq\
u_{C}(x)\). \(S(A,C)\leq S(A,B)\) and \(S(A,C)\leq S(B,C)\). We can get  
\[ \begin{array}{l}D(A,B)=\max\limits_{x\in U}\left\{\frac{1}{2}\left(\left|\mu _{A}^{2}(x)-\mu_{B}^{2}(x)\right|+\left|\
u_{A}^{2}(x)-\
u_{B}^{2}(x)\right| \right)\right\}\\ \leq\max\limits_{x\in U}\left\{\frac{1}{2}\left(\left|\mu_{A}^{2}(x)-\mu_{C}^ {2}(x)\right|+\left|\
u_{A}^{2}(x)-\
u_{C}^{2}(x)\right|\right)\right\}=D(A,C )\\ D(B,C)=\max\limits_{x\in U}\left\{\frac{1}{2}\left(\left|\mu_{A}^{2}(x)-\mu_{ C}^{2}(x)\right|+\left|\
u_{B}^{2}(x)-\
u_{C}^{2}(x)\right|\right)\right\}\\ \leq\max\limits_{x\in U}\left\{\frac{1}{2}\left(\left|\mu_{A}^{2}(x)-\mu_{C}^ {2}(x)\right|+\left|\
u_{A}^{2}(x)-\
u_{C}^{2}(x)\right|\right)\right\}=D(A,C )\end{array} \]  
Then \(D(A,C)\geq D(A,B)\) and \(D(A,C)\geq D(B,C)\). Namely, \(\sqrt{D(A,C)}\geq\sqrt{D(A,B)}\) and \(\sqrt{D(A,C)}\geq\sqrt{D(B,C)}\). Therefore, \(\vartheta(A,C)\subseteq\vartheta(A,B)\) and \(\vartheta(A,C)\subseteq\vartheta(B,C)\). \(\square\)  
Based on Theorem 3, we introduce the above Pythagorean fuzzy closeness measure into a Pythagorean fuzzy decision system.