## 3 Interval-Valued Complex Fuzzy Relations
\right\}\text{on an IVCIFS}.\\ \bar{\mathbb{E}}=\left\{ \begin{array}{l}\left(u,[0.07,0.12]e^{2[0.24,0.29] \pi},[0.56,0.66]e^{2[0.57,0.71]\pi}\right),\\ \left(v,[0.63,0.71]e^{2[0.03,0.15]\pi},[0.11,0.23]e^{2[0.73,0.82]\pi}\right),\\ \left(w,[0.37,0.46]e^{2[0.15,0.25]\pi},[0.32,0.43]e^{2[0.53,0.68]\pi}\right) \end{array} \right\}\text{, then the IVCI-composite-}\\ \bar{\mathbb{R}}_{1}\circ\bar{\mathbb{R}}_{2}=\left\{ \begin{array}{l}\left((u,u),[ 0.07,0.12]e^{2[0.15,0.25]\pi},[0.56,0.66]e^{2[0.63,0.74]\pi}\right),\\ \left((v,w),[0.37,0.46]e^{2[0.03,0.15]\pi},[0.32,0.43]e^{2[0.73,0.82]\pi}\right),\\ \left((w,u),[0.07,0.12]e^{2[0.15,0.25]\pi},[0.56,0.66]e^{2[0.57,0.71]\pi}\right),\end{array} \right\}\text{, then the IVCI-composite-}\\ \bar{\mathbb{R}}_{1}\circ\bar{\mathbb{R}}_{2}=\left\{ \begin{array}{l}\left((u,u),[ 0.07,0.12]e^{2[0.15,0.25]\pi},[0.56,0.66]e^{2[0.63,0.74]\pi}\right),\\ \left((v,w),[0.37,0.46]e^{2[0.03,0.15]\pi},[0.32,0.43]e^{2[0.73,0.82]\pi}\right),\\ \left((w,u),[0.07,0.12]e^{2[0.15,0.25]\pi},[0.56,0.66]e^{2[0.57,0.71]\pi}\right),\end{array} \right\}\text{,}\right\}\]  
**Theorem 1**. : _An IVCIFR \(\bar{\mathbb{R}}\) is an IVCI-symmetric-FR on an IVCIFS \(\bar{\mathbb{E}}\) iff \(\bar{\mathbb{R}}=\bar{\mathbb{R}}^{\mathbb{C}}\)._  
**Proof. ** Assume that \(\bar{\mathbb{R}}=\bar{\mathbb{R}}^{\mathbb{C}}\), then  
\((u,v)\in\bar{\mathbb{R}}\Rightarrow(v,u)\in\bar{\mathbb{R}}^{\mathbb{C}} \Rightarrow(v,u)\in\bar{\mathbb{R}}\). Thus, \(\bar{\mathbb{R}}\) is an IVCI-symmetric-FR on an IVCIFS \(\bar{\mathbb{E}}\). Conversely, suppose that \(\bar{\mathbb{R}}\) is an IVCI-symmetric-FR on an IVCIFS \(\bar{\mathbb{E}}\), then  
\((u,v)\in\bar{\mathbb{R}}\Rightarrow(v,u)\in\bar{\mathbb{R}}\). However, \((v,u)\in\bar{\mathbb{R}}^{\mathbb{C}}\Rightarrow\bar{\mathbb{R}}=\bar{\mathbb{R }}^{\mathbb{C}}\). \(\square\)  
**Theorem 2**. : _An IVCIFR \(\bar{\mathbb{R}}\) is an IVCI-transitive-FR on an IVCIFS \(\bar{\mathbb{E}}\) iff \(\bar{\mathbb{R}}\circ\bar{\mathbb{R}}\subseteq\bar{\mathbb{R}}\)._  
**Proof. ** Assume that \(\bar{\mathbb{R}}\) is an IVCI-transitive-FR on an IVCIFS \(\bar{\mathbb{E}}\). Let \((u,w)\in\bar{\mathbb{R}}\circ\bar{\mathbb{R}}\),  
Then, by the definition of IVCI-transitivity-FR,  
\((u,v)\in\bar{\mathbb{R}}\) and \((v,w)\in\bar{\mathbb{R}}\Rightarrow(u,w)\in\bar{\mathbb{R}}\Rightarrow\bar{ \mathbb{R}}\circ\bar{\mathbb{R}}\subseteq\bar{\mathbb{R}}\). Conversely assume that \(\bar{\mathbb{R}}\circ\bar{\mathbb{R}}\subseteq\bar{\mathbb{R}}\), then  
\(\text{for }(u,v)\in\bar{\mathbb{R}}\) and \((v,w)\in\bar{\mathbb{R}}\Rightarrow(u,w)\in\bar{\mathbb{R}}\circ\bar{\mathbb{R }}\subseteq\bar{\mathbb{R}}\Rightarrow(u,w)\in\bar{\mathbb{R}}\). Thus, \(\bar{\mathbb{R}}\) is an IVCI-transitive-FR on \(\bar{\mathbb{E}}\). \(\square\)  
**Theorem 3**. : _Suppose \(\bar{\mathbb{R}}\) is an IVCI-equivalence-FR on an IVCIFS \(\bar{\mathbb{E}}\), then \(\bar{\mathbb{R}}\circ\bar{\mathbb{R}}=\bar{\mathbb{R}}\)._  
**Proof. **: Assume that \((u,v)\in\bar{\mathbb{R}}\),  
then by the definition of IVCI-symmetric-FR,  
\((v,u)\in\bar{\mathbb{R}}\). Now, by using the definition of IVCI-transitive-FR,  
\((u,u)\in\bar{\mathbb{R}}\). However, by the definition of IVCI-composite-FR,  
\((u,u)\in\bar{\mathbb{R}}\circ\bar{\mathbb{R}}\). Thus,  
\[\bar{\mathbb{R}}\subseteq\bar{\mathbb{R}}\circ\bar{\mathbb{R}} \tag{1}\]  
Conversely, assume that \((u,v)\in\bar{\mathbb{R}}\circ\bar{\mathbb{R}}\), then \(\exists\ w\in\mathbb{S}\
i(u,w)\in\bar{\mathbb{R}}\) and \((w,v)\in\bar{\mathbb{R}}\). However, it is given that \(\bar{\mathbb{R}}\) is an IVCI-equivalence-FR on \(\bar{\mathbb{E}}\), so \(\mathbb{R}\) is also an IVCI-transitive-FR. Therefore,  
\[(u,v)\in\bar{\mathbb{R}}\Rightarrow\bar{\mathbb{R}}\circ\bar{\mathbb{R}} \subseteq\bar{\mathbb{R}} \tag{2}\]  
Thus, by (1) and (2),  
\(\bar{\mathbb{R}}\circ\bar{\mathbb{R}}=\bar{\mathbb{R}}\). \(\square\)  
**Theorem 4**.