## 2 Materials and Methods
### Methods
#### 2.2.3. Rule Induction with Information Systems  
An information system [12] is defined by the elements: \((U,\,A,\,V,f)\), where \(U\) is a universe, \(A\) is a set of attributes, \(V\) represents attributes domains: \(\mathrm{V}=\mathrm{df}\bigcup_{a}\mathrm{V}_{a}\), with nonempty domain \(V_{a}\) of the \(a\)-th attribute (\(a\in A\)) and \(f\) is the so called information function \(f\): \(U\times A\to V\), \(\forall_{x\in\mathrm{U},\,a\in A}\mathrm{f}(u,a)\in\mathrm{V}_{a}\). Important role in information systems plays the _indiscernibility binary relation_, which is an equivalence relation, defined over \(U\): \(\mathrm{IND}(\mathrm{B})=\mathrm{df}\{(x,y)\in U\times U\colon\forall_{a\in \mathrm{B}}\mathrm{f}(x,a)\!=\!\mathrm{f}(y,a)\}\), under which the _lower_ and _upper_ approximations of any subset of \(U\) can be defined, respectively:  
\[\mathrm{B}\!\downarrow\!\mathrm{X}=\mathrm{df}\{x\in\mathrm{U}\colon[x]_{ \mathrm{IND}}\subseteq X\}, \tag{6}\]  
\[\mathrm{B}\!\uparrow\!\mathrm{X}=\mathrm{df}\{x\in\mathrm{U}\colon[x]_{ \mathrm{IND}}\cap X\
eq\varnothing\},\,\mathrm{B}\subseteq A\text{ and }X\subseteq\mathrm{U}. \tag{7}\]  
The above approximations define a rough set [13; 14]. We can consider an information system as a decision table if a decision attribute is introduced. With this assumption, a decision making approach was introduced by [PERSON] and [PERSON] [41; 42], presenting a rule induction process with respect to all considered decisions. The procedure consists of the following steps:  
1. Introduce an information system with decision attribute,
2. Eliminate information system inconsistency, i.e., objects with the same information function values, but different decision values, applying _lower_ or _upper_ approximation precision analysis,
3. Provide attribute _reduct_,
4. Apply rule induction and define rules for all considered decisions, which cover the decision problem. Below, we explain the main algorithms applied in the procedure above with examples. The _reduct_ generation is omitted, as any other attribute selection method can be applied. Inconsistency Elimination  
Let consider the following information system with decision attribute: (U, A \(\cup\) {a*}, \(\mathrm{V}\cup\mathrm{V}_{a*}\), \(f\)). If inconsistency is found, then Algorithm 1, which uses the lower approximation precision as a decisive factor, should be applied. The lower approximation precision is defined as follows:  
\[\gamma_{\mathrm{B}}(\mathrm{X})\ =\mathrm{df}\ \frac{|\mathrm{B}\!\downarrow\! \mathrm{X}|}{|\mathrm{U}|},\,\mathrm{B}\subseteq A \tag{8}\]  
Figure 4: Combined FOU for two sample rules with two interval type-2 (IT2) sets in the conclusions. The final system output is defined as \((y_{1}+y_{\mathrm{r}})/2\), where \(y_{1}\) and \(y_{\mathrm{r}}\) are calculated with respect to \(L\) and \(R\) values, which can be determined using the Karnik–Mendel algorithm. where \(B\subseteq A\) and  
\[d^{B}_{a^{*}}(x_{k})\ =\ _{df}\ \{v\ \in\ V_{a^{*}}\colon\exists_{y\in U}(x_{k} \operatorname{IND}(B)\ y)\ \wedge\ f(y,a^{*})=\ v\} \tag{10}\]  
For simplicity, let consider the example below that illustrates the generation of matrix \(M_{1}\)  
**Example 2**. : _Let assume an information system with decision attribute extended by the corresponding \(d^{B}_{a^{*}}\) values (\(U=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}\), \(A=B=\{a,b,c\}\), \(V_{a}=V_{b}=V_{c}=V_{a^{*}}=\{0,1,2\}\)):_  
\[ \begin{array}{c|c|c|c|c}&a&b&c&a^{*}&d^{B}_{e^{*}}\\ x_{1}&1&0&1&0&\{0\}\\ x_{2}&0&0&0&1&\{1\}\\ x_{3}&2&0&1&0&\{0\}\\ x_{4}&0&0&1&2&\{2\}\\ x_{5}&1&1&1&0&\{0\}\\ \hline\end{array} \]  
_The corresponding discernibility matrix takes the form:_  
\[ \begin{array}{c|c|c|c|c|c}&x_{1}&x_{2}&x_{3}&x_{4}&x_{5}\\ x_{1}&\varnothing&&&&&\\ x_{2}&\{a,c\}&\varnothing&&\cdots&&\\ x_{3}&\{a\}&\{a,c\}&\varnothing&&\\ x_{4}&\{a\}&\{c\}&\{a\}&\varnothing&&\\ x_{5}&\{b\}&\{a,b,c\}&\{a,b\}&\{a,b\}&\varnothing\\ \hline\end{array} \]  
_The matrix \(M\)(IS) is a symmetric matrix.