## 5 Diagnostic Information Granules
### The Diagnostic Granule System: Its Language and Interpretation
\(p_{1}\subseteq_{G}p_{2}\) iff for any \(x\in X,\langle\mu_{p1}(x),v_{p1}(x)\rangle\leq_{n}\langle\mu_{p2}(x),v_{p2}(x)\rangle\), i.e., \(\mu_{p1}(x)\leq\mu_{p2}(x)\), and \(v_{p1}(x)\geq v_{p2}(x)\),  
11. \(p_{1}=_{G}p_{2}\) iff \(\cup_{G}\{z\in G_{0}:(z\subseteq_{G}p_{1})\Leftrightarrow(z\subseteq_{G}p_{2} )\}=1_{G}\). For any \(e\in G_{inst},p\in G_{set}\):  
12. \(e\in_{G}p\) iff \(\{e\}_{G}\subseteq_{G}p\). Symbols \(0_{G},1_{G}\), are interpreted by formulas:  
13. \(p=0_{G}\) iff for any \(x\in X,\langle\mu_{p}(x),v_{p}(x)\rangle=\langle 0,1\rangle\),  
14. \(p=1_{G}\) iff for any \(x\in X,\langle\mu_{p}(x),v_{p}(x)\rangle=\langle 1,0\rangle\). Similarly to the set theory, it can be assumed that:  
15. \(y^{G}=\cup_{G}\{t\in G_{0}:\
eg t\subseteq_{G}y\}\),  
16. \(y\setminus_{G}z=\cup_{G}\{t\in G_{0}:t\subseteq_{G}y\wedge\
eg t\subseteq_{G} z)\}\). By applying the above conditions (1)-(16), and simple calculus, Definition 9, Fact 5, and Theorem 11 are proved. **Fact 5**. :
1. \(\cup_{G}\varnothing=0_{G}\)_,_
2. \(\cup_{G}G_{0}=1_{G}\)_,_
3. \(y^{G}=1_{G}\)_,_
4. \(y\setminus_{G}1_{G}=0_{G}\)_,_
5. \(y\setminus_{G}z\subseteq_{G}y\cap_{G}z^{G}=y\)_._  
**Theorem 11**. : _For any \(p_{1},p_{2}\in G_{set}\) and \(A,B\subseteq G_{set}\):_  
1. \(p_{1}\subseteq_{G}p_{2}\) _iff for any_ \(t\in G_{0}\)_, if_ \(t\subseteq_{G}p_{1}\)_, then_ \(t\subseteq_{G}p_{2}\)_,_
2. \(p_{1}\subseteq_{G}p_{2}\) _iff for any_ \(e\in G_{inst}\)_, if_ \(x\in_{G}p_{1}\)_, then_ \(e\in_{G}p_{2}\)_,_
3. \(p_{1}=_{G}p_{2}\) _iff_ \(p_{1}\subseteq_{G}p_{2}\) _and_ \(p_{2}\subseteq_{G}p_{1}\)_,_
4. \(p_{1}=_{G}p_{2}\) _iff_ \(p_{1}=p_{2}\)_,_
5. \(p_{1}=\cup_{G}\{t\in G_{0}:t\subseteq_{G}p_{1}\}\)_,_
6. _if_ \(A\subseteq B\)_, then_ \(\cup_{G}A\subseteq_{G}\cup_{G}B\)_,_
7. _for any_ \(e\in G_{0}\) _exists_ \(p\in A\) _such that_ \(e\subseteq_{G}p\) _iff_ \(e\subseteq_{G}\cup_{G}A\)_._  
Proof. : For example, condition (5) of Theorem 11 is proven. Let \(e\in G_{0}\). If \(e\subseteq_{G}p_{1}\), then \(e\subseteq_{G}\cup_{G}\{t\in G_{0}:t\subseteq_{G}p_{1}\}\). If \(e\subseteq_{G}p=\cup_{G}\{t\in G_{0}:t\subseteq_{G}p_{1}\}\), then for any \(x\in X,e(x)=\langle u_{e}(x),v_{e}(x)\rangle\leq_{n}\langle\mu_{p}(x),v_{p}(x) \rangle=\sup_{n}\{\langle\mu_{t}(x),v_{t}(x)\rangle:t\in G_{0},t\subseteq_{G}p_ {1}\}\). Since \(\langle u_{e}(a),v_{e}(a)\rangle\leq_{n}\langle\mu_{p}(x),v_{p}(x)\rangle\leq_ {n}\langle\mu_{p1}(x),v_{p1}(x)\rangle\), so \(e\subseteq_{G}p_{1}\). Hence, based on condition (3) of Theorem 11, there is \(p_{1}=\cup_{G}\{t\in G_{0}:t\subseteq_{G}p_{1}\}\). Hence and based on Theorem 8, introducing a notation agreement: \(\Sigma_{G}^{\lambda}p=_{df}\lambda p\), \(\Pi_{G}^{\lambda}p=_{df}p^{\lambda}\), we get:  
**Theorem 12**. : _For any \(p_{1},p_{2},p_{3}\in n\)-ScIFN and any number \(\lambda,\lambda_{1},\lambda_{2}\in[0,1]\):_  
1. \(p_{1}\cup_{G}0_{G}=p_{1},p_{1}\cup_{G}1_{G}=1_{G}\)_,_
2. \(p_{1}\cup_{G}p_{2}=p_{2}\cup_{G}p_{1}\)_,_
3. \((p_{1}\cup_{G}p_{2})\cup_{G}p_{3}=p_{1}\cup_{G}(p_{2}\cup_{G}p_{3})\)_,_
4. \(p_{1}\cap_{G}1_{G}=p_{1},p_{1}\cap_{G}0_{G}=0_{G}\)_,_
5. \(p_{1}\cap_{G}p_{2}=p_{2}\cap_{G}p_{1}\)_,_
6. \((p_{1}\cap_{G}p_{2})\cap_{G}p_{3}=p_{1}\cap_{G}(p_{2}\cap_{G}p_{3})\)_,_
7. \(\lambda(p_{1}\cup_{G}p_{2})=\lambda p_{1}\cup_{G}\lambda p_{2}\)_,_
8. \((p_{1}\cap_{G}p_{2})^{\lambda}=p_{1}^{\lambda}\cap_{G}p_{2}^{\lambda}\)_,_
9. \((\lambda_{1}+\lambda_{2})p_{1}=\lambda_{1}p_{1}\cup_{G}\lambda_{2}p_{1}\)_,_
10. \(p_{1}^{\lambda_{1}+\lambda_{2}}=p_{1}^{\lambda_{1}}\cap_{G}p_{1}^{\lambda_{1}}\)_._  
**Definition 11**. : _Conditions (1)-(10) in Theorem 12 define the **granular algebra of diagnosis** defined on the n-SclFS set._