## 5 Diagnostic Information Granules
### Information granule System Database  
Using the constructions on rough sets given by [PERSON] in the papers [42; 44], it is possible to define the simplest sub-system of the granules system, in which from the set of instances \(G_{inst}\) using the operation \(\{\}_{G}\), transforming the instances as elementary granules (granular elements) into granular singletons and operation of singleton summation \(\cup_{G}\), any information granule is obtained. **Definition 12**. : _Any granule system \(\mathbf{G}\):_  
\[\mathbf{G_{base}}=\langle G,\{\}_{G},\cup_{G},\subseteq_{G},=_{G},0_{G},1_{G},G_{inst},G_{set},G_{0}\rangle, \tag{8}\]  
_is called the **information granule database**, when the elements of the \(G\) set are called granules, operations \(\{\}_{G}\) is the singleton, \(\cup_{G}\) is the granule sum, \(\subseteq_{G}\) is the inclusion relation, \(=_{G}\) is the closeness relation, \(0_{G}\) is the empty granule and \(1_{G}\) is the full granule, and \(G_{inst}\) is the set of instance granules, \(G_{set}\) is the set of set granules, and \(G_{0}\) is the singleton set of instance granules. The system \(\mathbf{G_{base}}\) satisfies the following axioms:_  
1. \(G_{0}\subseteq G_{set}\)_,_
2. \(G=G_{inst}\cup G_{set}\)_,_
3. \(G_{inst}\cap G_{set}=\varnothing\)_._  
_For any_ \(x\in G_{set}\)_:_
4. \(x\subseteq_{G}x\)_,_
5. \(0_{G}\subseteq_{G}x\)_,_
6. _If_ \(x\
eq 0_{G}\)_, then it is not true that_ \(x\subseteq_{G}0_{G}\)_._  
_There is some function \(\{\}_{G}:G_{inst}\to 0_{0}\), such that:_  
1. _For any_ \(x\in G_{inst}\)_,_ \(\{x\}_{G}\subseteq_{G}1_{G}\)_,_
2. \(G_{0}=\{x\in G_{0}:\exists z\in G_{inst}(x=\{z\}_{G})\}\)_._  
_There is some function \(\cup_{G}:\varphi(G_{0})\to G_{set}\), such that:_  
1. \(\cup_{G}\varnothing=0_{G}\)_,_
2. \(\cup_{G}\mathbb{G}=0_{G}\)_,_
3. \(\cup_{G}\mathbb{G}=1_{G}\)_,_
4. _for any_ \(y\in G_{set},y=\cup_{G}\{x\in G_{0}:x\subseteq_{G}y\}\)_,_
5. _for any_ \(x,y\in G_{set}\)_,_ \(x=_{G}y)\(\Leftrightarrow_{G}\cup_{G}\{z\in G_{0}:(z\subseteq_{G}x)\Leftrightarrow( z\subseteq_{G}y)\}=1_{G}\)_._  
The information granule database \(\mathbf{G_{base}}\) is a diagnostic granule subsystem. When we join to the conditions (1)-(12), a Definition: _for any \(x\in G_{inst}\) and \(y\in G_{set}\)_, \(x\in_{G}y\)_iff \(\{x\}_{G}\subseteq_{G}y\)_._  
Then we get the following Theorem. **Theorem 13**. :
1. \(\{x\in G_{inst}:x\in_{G}0_{G}\}=\varnothing\)_,_
2. \(\{x\in G_{inst}:\exists z\in G_{0}(x\in_{G}z)\}=G_{inst}\)_,_
3. \(\{z\in G_{set}:\exists x\in G_{inst}(x\in_{G}z)\}=G_{set}\)_,_
4. \(y=\cup_{G}\{\{x\}_{G}:x\in_{G}y\}\)_, for any_ \(y\in G_{set}\)_._  
Theorem 7 describes the set granules analogously to the classical set theory. The following theorem answers question about the analogy of operation on diagnostic granules to set operations. **Theorem 14**. : _In the granule database \(\mathbf{G_{diag}}\), the diagnostic granules \(y^{\prime G},y\cap_{G}z,y\cup_{G}z,y\setminus_{G}z\), are described by formulas:_  
1. \(y^{\prime G}=\cup_{G}\{z\in G_{0}:\exists x\in G_{inst}(x\in_{G}z\wedge\
eg x \in_{G}y)\}\)_,_\(y\cap_{G}z\subseteq_{G}\cup_{G}\{t\in G_{0}:\exists x\in G_{inst}(x\in_{G}t\wedge x \in_{G}y\wedge x\in_{G}z)\}\), \(\cup_{G}\{t\in G_{0}:\exists x\in G_{inst}(x\in_{G}t\wedge(x\in_{G}y\lor x\in_{G} z))\}\subseteq_{G}y\cup_{G}z\), \(y\setminus_{G}z=\cup_{G}\{t\in G_{0}:\exists x\in G_{inst}(x\in_{G}t\wedge x \in_{G}y\wedge\
eg x\in_{G}z)\}\). Proof. : Conditions (1) and (4) are fulfilled by axioms (15) and (16) of Definition 10 of the diagnostic granule system and of Fact 5. Conditions (2) and (3) are proofed based on Theorems 10 and 11 and Definition 10. Proof of condition (2). \(y\cap_{G}z\subseteq_{G}y\) and \(y\cap_{G}z\subseteq_{G}z\). Hence \(\{t\in G_{0}:t\subseteq_{G}y\cap_{G}z\}\subseteq\{t\in G_{0}:t\subseteq_{G}y \wedge t\subseteq_{G}z)\}=\{t\in G_{0}:\exists x\in G_{inst}(x\in_{G}t\wedge x \in_{G}y\wedge x\in_{G}z)\}\).