# An Intuitionistic Fuzzy-Rough Set-Based Classification for Anomaly Detection
## 3 Problem Definitions  
Below, we present some important terms and definitions used in the paper.  
**Definition 1**.: _Fuzzy set._  
Let \(X=\left\{x_{1},\,x_{2},\,\ldots\,x_{n}\right\}\) be the universe of discourse. A fuzzy set [32]\(A\) on \(X\) is characterized by  
\[A=\left\{(x_{i},\mu_{A}(x_{i}));x_{i}\in X,i=1,2,\ldots n\right\} \tag{1}\]  
where \(\mu_{A}:X\rightarrow\left[0,1\right],\) the membership function, gives the grade of membership of each element \(x_{i}\in X\) in \(A\).  
**Definition 2**.: _Intuitionistic fuzzy set._  
[PERSON] [33] proposed the definition of an intuitionistic fuzzy set \(A\) on \(X\) as  
\[A=\left\{(x_{i},\mu_{A}(x_{i}),\
u_{A}(x_{i}));x_{i}\in X,i=1,2,\ldots n\right\} \tag{2}\]  
where \(\mu_{A}:X\rightarrow\left[0,1\right]\) and \(\
u_{A}:X\rightarrow\left[0,1\right]\) are the membership function and nonmembership function of the fuzzy set \(A\), respectively, satisfying the condition \(0\leq\mu_{A}(x_{i})+\
u_{A}(x_{i})\leq 1\), for every \(x_{i}\in X\).  
**Definition 3**.: _Correlation of intuitionistic fuzzy sets._  
Let \(A=\left\{(x_{i};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\  
For any data instances \(x_{i}\); \(i=1,2,\ldots\ m\) in \(U\) and \(0<\alpha\leq 1\), the \(\alpha\)--cut \(R_{\alpha}\) of \(R\) generates \(\alpha\)-relation (\(U,\rho\)) as  
\[\alpha(x_{i})=\{x;\ \rho_{x_{i}x}\geq\ \alpha\}. \tag{5}\]  
Proposition 1: [37; 38]  
If a fuzzy relation \(R\) is an equivalence relation in max-min sense, then for \(\alpha\in(0,1)\), \(R_{\alpha}\) possesses an equivalence relation. Therefore, any \(\alpha\)--relation represented by an \(\alpha\)--cut \(R_{\alpha}\) will have an equivalence relation. The ordered pair (\(U,R_{\alpha}\)) is an approximation space.  
Definition 7: Fuzzy-Rough Set.  
Fuzzy-rough set theory is an extension of rough set theory where the crisp equivalence class concept is extended to form fuzzy equivalence classes. Let the conditional and decision attributes of an information systems both be intuitionistic fuzzy sets and let us define an \(\alpha\)--relation in the aforesaid manner. Since a fuzzy equivalence relation generates a fuzzy partition of the universe of discourse, the \(\alpha\)--relation will generate a series of fuzzy equivalence classes [49; 50; 51], known as fuzzy knowledge granules. Letting (\(U,R\)) represent a fuzzy approximation space and \(X\) be a fuzzy subject of \(U\) intuitionistic sense, the intuitionistic fuzzy nano lower approximation, the intuitionistic fuzzy nano upper approximation, and the intuitionistic nano boundary approximation of \(X\) on (\(U,R\)) are denoted by \(\underline{I}(X)\), \(\overline{I}(X)\), and \(B_{I}(X)\), respectively, which are expressed as follows [48]:  
\[\underline{I}(X)=\big{\{}\big{(}\kappa,\mu_{\underline{R}X}(x),\
u_{\underline {R}X}(x)\big{)},y\in[x]_{R},x\in U\big{\}} \tag{6}\]  
\[\overline{I}(X)=\bigg{\{}\bigg{(}\kappa,\mu_{\underline{R}X}(x),\
u_{ \underline{R}X}(x)\bigg{)},y\in[x]_{R},x\in U\bigg{\}} \tag{7}\]  
\[B_{I}(X)=\overline{I}(X)-\underline{I}(X) \tag{8}\]  
where \(\mu_{\underline{R}X}(x)=\inf_{y\in[x]_{R}}(y)\), \(\
u_{\underline{R}X}(x)=sup_{y\in[x]_{R}}(y)\), \(\mu_{\underline{R}X}(x)=sup_{y\in[x]_{R}}(y)\), and \(\
u_{\underline{R}X}(x)=inf_{y\in[x]_{R}}(y)\).