## 2 Materials and Methods
### Extension Sanchez's Method (Composite Fuzzy Relation 1, Cft1)  
The extension of [PERSON]'s method (DBR) [29] for medical diagnosis is the major way to extract a composite fuzzy relation and it is very useful in medical diagnosis.  
Let two non-empty sets be \(p_{1}\) where \(l\in\{1,\ldots,L\}\), and \(s_{l}\), where \(j\in\{1,\ldots,n\}\). A fuzzy relation \(Q\) from \(p_{1}\) to \(s_{l}\) is a fuzzy set of \(P\times S\), characterized by \(\mu_{Q}\) and \(\
u_{Q}\)[5].  
Let two non-empty sets be \(s_{l}\) where \(j\in\{1,\ldots,n\}\), and \(d_{m}\) where \(m\in\{1,\ldots,M\}\). A fuzzy relation \(R\) from \(s_{l}\) to \(d_{m}\) is a fuzzy set of \(S\times D\), characterized by \(\mu_{R}\) and \(\
u_{R}\)[5].  
Figure 1: Presentation of functions pairs for \(q=1,q=2,q=3,q=10\).  
Let \(Q(P\longrightarrow S)\) and \(R(S\longrightarrow D)\) be two fuzzy relations. We define \(T=R\circ Q\), \(T\in(P\times D)\), and the-composite fuzzy relation of \(R\) and \(Q\), by \((R\circ Q)(p,d)=\bigvee_{s}[Q(p,s)\wedge R(s,d)]\) or  
\[\mu_{T}(p_{I},d_{m})=\bigvee_{s\in S}\left[\mu_{Q}(P_{m},s)\ \bigwedge\mu_{R} \big{(}s_{j},d_{m}\big{)}\right] \tag{14}\]  
where \(s\in S\), for all \((p,d)\in P\times D\), \(\bigvee=max\) and \(\bigwedge=min\), \(p_{I}=patient\) for \(l=1\ldots L\), \(d_{m}=each\) disease, and \(s_{j}=\) symptoms [11; 30].  
The Equation (14) is called max-min composition and calculates the membership function after the interaction of \(Q(P\longrightarrow S)\) which links patients with symptoms, and \(R(S\longrightarrow D)\) which links symptoms with disease [4; 10; 11].  
The non-membership function we used was the \(\alpha\)-composite fuzzy relation. Let \(Q(P\longrightarrow S)\) and \(R(S\longrightarrow D)\) be two fuzzy relations. We defined \(T=R\circ Q\) and \(T\in(P\times D)\) by \((R\circ Q)(p,d)=\bigwedge_{s}[Q(p,s)\propto R(s,d)]\) or  
\[\
u_{T}(p_{I},d_{m})=\bigwedge_{s\in S}\left[\
u_{Q}(p_{I},s_{j})\ \bigvee\
u_{R} \big{(}s_{j},d_{m}\big{)}\right] \tag{15}\]  
where \(s\in S\), for all \((p,d)\in P\times D\), \(\bigvee=max\) and \(\bigwedge=min\), \(p_{I}=patient\) for \(l=1\ldots L\), \(d_{m}=each\) disease, and \(s_{j}=\) symptoms [11; 30].  
The Equation (15) is called min-max composition and calculates the non-membership function after the interaction of \(Q(P\longrightarrow S)\) which links patients with symptoms, and \(R(S\longrightarrow D)\) which links symptoms with disease [4; 10; 11].  
Composite Fuzzy Relation 2 (CFR2)  
There is another way to calculate the composition between two fuzzy relations. Let \(Q(P\longrightarrow S)\) and \(R(S\longrightarrow D)\) be two fuzzy relations. We defined \(T=R\circ Q\), \(T\in(P\times D)\), and the max-average composition of \(R\) and \(Q\) by the membership function:  
\[\mu_{T}(p_{I},d_{m})=\bigvee_{s\in S}\left[\frac{\mu_{Q}\big{(}p_{I},s_{j} \big{)}+\mu_{R}\big{(}s_{j},d_{m}\big{)}}{2}\right] \tag{16}\]  
where \(s\in S\), for all \((p,d)\in P\times D\), \(\bigvee=max\), \(p_{I}=patient\) for \(l=1\ldots L\), \(d_{m}=each\) disease, and \(s_{j}=\) symptoms [5].  
For non-membership functions we used the min-average composition of \(R\) and \(Q\):  
\[\
u_{T}(p_{I},d_{m})=\bigwedge_{s\in S}\left[\frac{\mu_{Q}\big{(}p_{I},s_{j} \big{)}+\mu_{R}\big{(}s_{j},d_{m}\big{)}}{2}\right] \tag{17}\]  
where \(s\in S\), for all \((p,d)\in P\times D\), \(\bigwedge=min\), \(p_{I}=patient\) for \(l=1\ldots L\), \(d_{m}=each\) disease, and \(s_{j}=\) symptoms [5].