## 6 An Example of a Standard Diagnostic granule System
### Examples of Calculations on Granules, Their Order, and Elements  
On the basis of axioms (6) and (7) of Definition 10 and conditions **S2** and **S3** in the standard diagnostic granules system, the following calculation formulas are adopted for the diagnostic generators \(p_{1},p_{2}\):  
**S2a.**: \((p_{1}\cup_{G}p_{2})(o)=(p_{1}\oplus p_{2})(o)=\langle\mu_{p1}(o),v_{p1}(o) \rangle\oplus\langle\mu_{p2}(o),v_{p2}(o)\rangle=\langle\max\{u_{p1}(o),u_{p2}( o)\},\min\{v_{p1}(o),v_{p2}(o)\}\rangle\),
**S3a.**: \((p_{1}\cap_{G}p_{2})(o)=(p_{1}\otimes p_{2})(o)=\langle\mu_{p1}(o),v_{p1}(o) \rangle\otimes\langle\mu_{p2}(o),v_{p2}(o)\rangle=\langle\min\{u_{p1}(o),u_{p2 }(o)\},\max\{v_{p1}(o),v_{p2}(o)\}\rangle\).  
The calculation results are presented in Table 2.  
According to Table 2, \(p_{1}\cup_{G}p_{2}=p_{2}\) and \(p_{1}\cap_{G}p_{2}=p_{1}\). This is also due to the fact that \(p_{1}\subseteq_{G}p_{2}\), i.e., \(p_{1}(o)\leq_{n}p_{2}(o)\) iff \(\langle\mu_{p1}(o),v_{p1}(o)\rangle\leq_{n}\langle\mu_{p2}(o),v_{p2}(o)\rangle\) iff \(\mu_{p1}(o)\leq\mu_{p2}(o)\) and \(v_{p2}(o)\leq v_{p1}(o)\).  
For example, \(p_{1}(4)\leq_{n}p_{2}(4)\) iff \(\langle\mu_{p1}(4),v_{p1}(4)\rangle\leq_{n}\langle\mu_{p2}(4),v_{p2}(4)\rangle\) iff \(\langle 0.5,0.9\rangle\leq_{n}\langle 0.8,0.6\rangle\) iff \(0.5\leq 0.8\) and \(0.6\leq 0.9\).  
Indeed, if \(\mu_{p1}(o)\leq\mu_{p2}(o)\) and \(v_{p2}(o)\leq v_{p1}(o)\), then from Formulas (**S2a**) and (**S3a**) we get:  
* \((p_{1}\cup_{G}p_{2})(o)=\langle\max\{u_{p1}(o),u_{p2}(o)\},\min\{v_{p1}(o),v_{ p2}(o)\}\rangle=\langle\mu_{p2}(o),v_{p2}(o)\rangle=p_{2}(o)\),
* \((p_{1}\cap_{G}p_{2})(o)=\langle\min\{u_{p1}(o),u_{p2}(o)\},\max\{v_{p1}(o),v_{ p2}(o)\}\rangle=\langle\mu_{p1}(o),v_{p1}(o)\rangle=p_{1}(o)\).  
The above formulas can also be analyzed in the context of determining the singletons of granules contained in the generators of granules or by examining the elements of these generators; Table 3 shows selected singletons contained in generator \(p_{1}\).  
If \(p_{1}(o_{i})=\langle\mu_{p1}(o_{i}),v_{p1}(o_{i})\rangle\
eq\langle 0,1\rangle\), then \(e(o_{i})=\langle\mu_{e}(o_{i}),v_{e}(o_{i})\rangle\leq_{n}\langle\mu_{p1}(o _{i}),v_{p1}(o_{i})\rangle\) and \(e(o_{i})=\langle 0,1\rangle\) for \(o\
eq o_{i}\).  
Thus \(e\) is a singleton contained in \(p\): \(e\in G_{0},e\subseteq_{G}p_{1}\).  
Hence \(\langle o_{i},\langle\mu_{e}(o_{i}),v_{e}(o_{i})\rangle\rangle\in_{G}p_{1}\).  
For example: \(\langle 3,\langle\mu_{\varnothing}(3),v_{\varnothing}(3)\rangle\rangle=\langle 3,\langle 0.7,0.7\rangle\rangle\in_{G}p_{1}(3)=\langle 0.8,0.4\rangle\). Moreover \(e_{2}\subseteq_{G}e_{1}\).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Object** & \(p_{1}(o)\) & \(p_{2}(o)\) & \((p_{1}\oplus p_{2})(o)\) & \((p_{1}\oplus p_{2})(o)\) & \((p_{1}\otimes p_{2})(o)\) \\ \hline
**O** & \(s\) & \(f\) & \(s\) & \(f\) & \(s\) & \(f\) & \(s\) & \(f\) \\ \hline
1 & 0.5 & 0.4 & 0.7 & 0.4 & 0.7 & 0.4 & 0.5 & 0.4 \\
2 & 0.8 & 0.8 & 0.9 & 0.4 & 0.9 & 0.4 & 0.8 & 0.8 \\
3 & 0.8 & 0.4 & 0.9 & 0.4 & 0.9 & 0.4 & 0.8 & 0.4 \\
4 & 0.5 & 0.9 & 0.8 & 0.6 & 0.8 & 0.6 & 0.5 & 0.9 \\
5 & 0.4 & 0.9 & 0.8 & 0.8 & 0.8 & 0.8 & 0.4 & 0.9 \\ \hline \end{tabular}
\end{table}
Table 2: Calculation of \((p_{1}\oplus p_{2})(o)=\langle s,f\rangle\) and \((p_{1}\otimes p_{2})(o)=\langle s,f\rangle\) for the diagnostic generators \(p_{1},p_{2}=\langle s,f\rangle\).