## 6 An Example of a Standard Diagnostic granule System
### Calculation of the Repeatability of Diagnostic Granules  
Repeatability of diagnostic granules in the map scale  
\(Sc_{n}(x)=x/n,Sc_{n}^{-1}(x)=nx\), for \(n>0,x\in[0,1],\langle s,f\rangle\in n\)-_SclFS_ (Example 3). \(\lambda\langle s,f\rangle=\langle Sc_{n}^{-1}(\lambda Sc_{n}(s)),Sc_{n}^{-1}(1- \lambda(1-Sc_{n}(f)))\rangle=\langle n\lambda s/n,n(1-\lambda(1-f/n))\rangle= \langle\lambda s,n-n\lambda+\lambda f\rangle\),  
\(\langle s,f\rangle^{\lambda}=\langle Sc_{n}^{-1}(1-\lambda(1-Sc_{n}(s))),Sc_{n} ^{-1}(\lambda Sc_{n}(f))\rangle=\langle n(1-\lambda(1-fsn)),n\lambda s/n\rangle= \langle n-n\lambda+\lambda s,\lambda f\rangle\). Repeatability of diagnostic granules in the logarithmic scale. \(Sc_{n}(x)=1-\log_{(1-1/n)}x,Sc_{n}^{-1}(x)=x\) and for \(n>1,Sc_{n}^{-1}(x)=(1-1/n)\)\(\tilde{\ }x\), for \(n>1,x\in(0,1]\), when \(Sc_{1}(x)=x\) and \(Sc_{n}(0)=0,\langle s,f\rangle\in n\)-_SclFS_ (Example 4), where \(a\)\(\tilde{\ }x=_{df}\)\(\tilde{\ }x\),  
\(\lambda\langle s,f\rangle=\langle Sc_{n}^{-1}(\lambda Sc_{n}(s)),Sc_{n}^{-1}(1- \lambda(1-Sc_{n}(f)))\rangle=\langle(1-1/n)\)\(\tilde{\ }\tilde{\ }\lambda(1-\log_{(1-1/n)}s),\)  
\((1-1/n)\)\(\tilde{\ }\tilde{\ }(1-\lambda(1^{\mbox{\tiny T}}1+\log_{(1-1/n)}f))\rangle= \langle((1-1/n)\)\(\tilde{\ }\tilde{\ }\lambda)/s\)\(\tilde{\ }\lambda,(1-1/n)\)\(\tilde{\ }\tilde{\ }\lambda(1-1/n)\)\(\tilde{\ }\tilde{\ }\lambda(1-1/n)\)\(\tilde{\ }\tilde{\ }\lambda(1-1/n)\)\(\tilde{\ }\tilde{\ }\lambda(1-1/n)/f\)\(\tilde{\ }\tilde{\ }\lambda))\)  
\(\langle s,f\rangle^{\lambda}=\langle Sc_{n}^{-1}(1-\lambda(1-Sc_{n}(s))),Sc_{n} ^{-1}(\lambda Sc_{n}(f))\rangle=\langle(1-1/n)/(s\)\(\tilde{\ }\tilde{\ }\lambda),((1-1/n)/f)\)\(\tilde{\ }\lambda)\rangle\). Repeatability of diagnostic granules in the power scale  
\(Sc_{n}(x)=x^{n},Sc_{n}^{-1}(x)=x^{1/n}\), for \(n>0,x\in[0,1],\langle s,f\rangle\in n\)-_SclFS_ (Example 5). \(\lambda\langle s,f\rangle=\langle Sc_{n}^{-1}(\lambda Sc_{n}(s)),Sc_{n}^{-1}(1- \lambda(1-Sc_{n}(f)))\rangle=\langle(\lambda s^{n})^{1/n},(1-\lambda(1-f^{n}) )^{1/n}\rangle\)  
\(=\langle\lambda^{1/n}s,(1-\lambda(1-f^{n}))^{1/n}\rangle\),  
\(\langle s,f\rangle^{\lambda}=\langle Sc_{n}^{-1}(1-\lambda(1-Sc_{n}(s))),Sc_{n} ^{-1}(\lambda Sc_{n}(f))\rangle=\langle(1-\lambda(1-s^{n}))^{1/n},\lambda^{1/ n}f\rangle\). It is easy to see that for \(\lambda=1,\lambda\langle s,f\rangle=\langle s,f\rangle^{\lambda}=\langle s,f\rangle\) and for \(\lambda=0,\lambda\langle s,f\rangle=\langle 0,1\rangle,\langle s,f\rangle^{ \lambda}=(1,0)\). Furthermore, in Tables 4-7, it is assumed that the diagnosis can be repeated a maximum of 5 times until certainty is achieved, i.e., \(\lambda=1-i/5\), where \(i\) is the number of repetitions, i.e., \(\lambda=1,4/5,3/5,2/5,1/5,0\). Tables 4 and 5 present the results of the calculation of the linear repeatability \(\lambda p_{1},\lambda p_{2}\) in the power scale \(Sc_{n}(x)=x^{n}\):  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multicolumn{12}{c}{**\(\lambda\)**} \\ \hline \multicolumn{12}{c}{**Object**} & \multicolumn{2}{c}{\(p_{1}\)} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**0.8**} & \multicolumn{2}{c}{**0.6**} & \multicolumn{2}{c}{**0.4**} & \multicolumn{2}{c}{**0.2**} & \multicolumn{2}{c}{**0**} \\ \hline \(o\) & \(s\) & \(f\) & \(s\) & \(f\) & \(s\) & \(f\) & \(s\) & \(f\) & \(s\) & \(f\) & \(s\) & \(f\) & \(s\) & \(f\) \\ \hline \(1\) & 0.50 & 0.40 & 0.50 & 0.40 & 0.47 & 0.69 & 0.44 & 0.80 & 0.40 & 0.8* A decrease in the repeatability degree \(\lambda\) leads to a decrease of the symptom degree \(s\),
* A decrease in the repeatability degree \(\lambda\) leads to an increase of the faults degree \(f\),
* \(\lambda p_{1}\subseteq_{G}\lambda p_{2}\) according to the condition (5) of Theorem 10, because \(p_{1}\subseteq_{G}p_{2}\).