## 6 An Example of a Standard Diagnostic granule System  
### Standard Diagnostic Granule System  
For any \(\langle\mu_{1},\
u_{1}\rangle\), \(\langle\mu_{2},\
u_{2}\rangle\in\textbf{n-ScIFS},\mu\in[0,1]\) there are the following formulas:  
**S1. **\(\mu_{1}\bullet_{s}\
u_{1}=\max\{\mu_{1},\
u_{1}\}\),  
\(\mu_{1}\bullet_{t}v_{1}=\min\{\mu_{1},\
u_{1}\}\),  
and then \(f_{s}(x)=x,f_{t}(x)=1-x\);  
\(\lambda\bullet_{t}\mu=f_{s}^{-1}[\lambda f_{s}(\mu)]=\lambda\mu\) and  
\(\lambda\bullet_{p}\mu=f_{t}^{-1}[\lambda f_{t}\mu)]=1-\lambda(1-\mu)\);  
**S2. **\(\langle\mu_{1},\
u_{1}\rangle\oplus\langle\mu_{2},\
u_{2}\rangle=\)  
\(\langle Sc_{n}^{-1}(Sc_{n}(\mu_{1})\bullet_{s}Sc_{n}(\mu_{2})),Sc_{n}^{-1}(Sc_{ n}(\
u_{1})\bullet_{t}Sc_{n}(\
u_{2}))\rangle=\)  
\(\langle\max\{u_{1},u_{2}\},\min\{v_{1},v_{2}\}\rangle\),  
**S3. **\(\langle\mu_{1},\
u_{1}\rangle\otimes\langle\mu_{2},\
u_{2}\rangle=\)  
\(\langle Sc_{n}^{-1}(Sc_{n}(\mu_{1})\bullet_{t}Sc_{n}(\mu_{2})),Sc_{n}^{-1}(Sc_{ n}(\
u_{1})\bullet_{s}Sc_{n}(\
u_{2}))\rangle=\)  
\(\langle\min\{u_{1},u_{2}\},\max\{v_{1},v_{2}\}\rangle\),  
**S4. **\(\lambda\langle\mu_{1},\
u_{1}\rangle=\)  
\(\langle Sc_{n}^{-1}(\lambda\bullet_{t}Sc_{n}(\mu_{1})),Sc_{n}^{-1}(\lambda \bullet_{p}Sc_{n}(\
u_{1}))\rangle=\)  
\(\langle Sc_{n}^{-1}(ASC_{n}(\mu_{1})),Sc_{n}^{-1}(1-\lambda(1-Sc_{n}(\
u_{1})))\rangle\),  
**S5. **\(\langle\mu_{1},\
u_{1}\rangle\lambda=\)  
\(\langle Sc_{n}^{-1}(\lambda\bullet_{p}Sc_{n}(\mu_{1})),Sc_{n}^{-1}(\lambda \bullet_{t}Sc_{n}(\
u_{1}))\rangle=\)  
\(\langle Sc_{n}^{-1}(1-\lambda(1-Sc_{n}(\mu_{1}))),Sc_{n}^{-1}(ASC_{n}(\
u_{1}))\rangle\). It is assumed that:  
* Symptom-fault diagnosis \((s,f)\) corresponds to the diagnostic granule \(p=(\mu_{p},v_{p})\in\textbf{n-ScIFS}\) defined by fuzzy sets \(\mu_{p}:X\rightarrow[0,1],v_{p}:X\rightarrow[0,1]\) and the values of the function \(\mu_{i},v_{j}\) in the scale \(Sc_{n}\) satisfy the condition for any \(x\in X\): \[0\leq Sc_{n}(\mu_{p}(x))+Sc_{n}(v_{p}(x))\leq 1.\] (9)
* For any diagnostic granule \(p=(\mu_{p},v_{p})\), granules \(\lambda p\) is a result of parallel repetition of an alternative of the diagnosis \(p\), and \(p^{\lambda}\) is the result of linear repetition of the conjunction of diagnosis \(p\). This is due to the inaccuracy, uncertainty, or imprecision of the diagnosis results. The degree of diagnosis that requires repetitions is the number \(\lambda\in(0,1]\). If the diagnosis does not need to be repeated and is certain, then \(\lambda=1\). The diagnosis can be repeated multiple times until it is certain. For example, if we want to repeat diagnosis maximally 5 times, then \(\lambda=1-i/5\), where \(i\) is the number of repetitions, then \(\lambda=1,4/5,3/5,2/5,1/5,0\). The diagnostic granule system determined by the assumptions described above will be called the **standard diagnostic granule system**. In the standard diagnostic granules system with the scaling function \(Sc_{n}\), for any granules \(y,z\in n\)-SclFS, the following conditions apply:  
1. \(y\cap_{G}z=_{df}\cup_{G}\{t\in C_{0}:\exists x\in G_{inst}(x\in_{G}t\wedge x \in_{G}y\wedge x\in_{G}z)\}\),
2. \(y\cup_{G}z=_{df}\cup_{G}\{y,z\}=\cup_{G}\{t\in C_{0}:\exists x\in G_{inst}(x \in_{G}t\wedge(x\in_{G}y\lor x\in_{G}z))\}\). The proof of the point (1):  
\(x=\langle a,\langle\mu_{p}(a),v_{p1}(a)\rangle\rangle\in_{G}\cup_{G}\{t\in G_ {0}:\exists x\in G_{inst}(x\in_{G}t\wedge x\in_{G}y\wedge x\in_{G}z)\}\) iff \(\langle\mu_{p}(a),v_{p}(a)\rangle\leq_{n}\langle\mu_{y}(a),v_{y}(a)\rangle \wedge\langle\mu_{p}(a),v_{p}(a)\rangle\leq_{n}\langle\mu_{z}(a),v_{z}(a)\rangle\) iff \(\mu_{p}(a)\leq\mu_{y}(a)\wedge v_{y}(a)\leq v_{p}(a)\wedge\mu_{p}(a)\leq\mu_{ z}(a)\wedge v_{z}(a)\leq v_{p}(a)\).