## 2 Preliminaries
\begin{array}{c}\Big{(}w,[0.28,0.36]e^{i2[0.43,0.61] \pi}\Big{)},\Big{(}x,[0.17,0.23]e^{i2[0.67,0.81]\pi}\Big{)},\\ \Big{(}y,[0.45,0.50]e^{i2[0.04,0.21]\pi}\Big{)},\Big{(}z,[0.89,0.98]e^{i2[0.016] \pi}\Big{)}\end{array} \right\}\) _is an IVCFS._  
**Definition 5.**_Ref. [16] Let \(\hat{\mathbb{E}}=\left\{\Bigg{(}x,\Big{[}\alpha_{\hat{\mathbb{E}}}^{-}(x), \alpha_{\hat{\mathbb{E}}}^{+}(x)\Big{]}e^{i2[\rho_{\hat{\mathbb{E}}}^{-}(x), \rho_{\hat{\mathbb{E}}}^{+}(x)]\pi}\Bigg{)}:x\in T\right\}\) and \(\hat{\mathbb{F}}=\left\{\Big{(}y,\Big{[}\alpha_{\hat{\mathbb{F}}}^{-}(y), \alpha_{\hat{\mathbb{F}}}^{+}(y)\Big{]}e^{i2[\rho_{\hat{\mathbb{F}}}^{-}(y), \rho_{\hat{\mathbb{F}}}^{+}(y)]\pi}\Big{)}:y\in T\right\}\) be two IVCFSs in a universe \(T\), then their Cartesian product is given as_  
\[\hat{\mathbb{E}}\times\hat{\mathbb{F}}=\Bigg{\{}\Bigg{(}(x,y),\Big{[}\alpha_{ \hat{\mathbb{E}}\times\hat{\mathbb{F}}}^{-}(x,y),\alpha_{\hat{\mathbb{E}}\times \hat{\mathbb{F}}}^{+}(x,y)\Big{]}e^{i2[\rho_{\hat{\mathbb{E}}}^{-}(x),\rho_{ \hat{\mathbb{E}}}^{+}(x,y)]\pi}\Bigg{)}:x\in\hat{\mathbb{E}},y\in\hat{\mathbb{ F}}\Bigg{\}}\]  
_where \(\alpha_{\hat{\mathbb{E}}\times\hat{\mathbb{F}}}^{-}(x,y)=\min\bigg{\{}\alpha_{ \hat{\mathbb{E}}}^{-}(x),\alpha_{\hat{\mathbb{F}}}^{-}(y)\bigg{\}}\), \(\alpha_{\hat{\mathbb{E}}\times\hat{\mathbb{F}}}^{+}(x,y)=\min\bigg{\{}\alpha_{ \hat{\mathbb{E}}}^{+}(x),\alpha_{\hat{\mathbb{F}}}^{+}(y)\bigg{\}}\rho_{\hat{ \mathbb{E}}\times\hat{\mathbb{F}}}^{-}(x,y)=\min\bigg{\{}\rho_{\hat{\mathbb{E} }}^{-}(x),\rho_{\hat{\mathbb{F}}}^{-}(y)\bigg{\}}\) and \(\rho_{\hat{\mathbb{E}}\times\hat{\mathbb{F}}}^{+}(x,y)=\min\bigg{\{}\rho_{ \hat{\mathbb{E}}}^{+}(x),\rho_{\hat{\mathbb{F}}}^{+}(y)\bigg{\}}\)._  
**Definition 6.**_Ref. [16] The interval-valued complex fuzzy relation (IVCFR) is a subset of the Cartesian product of any two IVCFSs, i.e., \(\hat{\mathbb{R}}\subseteq\hat{\mathbb{F}}\times\hat{\mathbb{F}}\), where \(\hat{\mathbb{E}}\) and \(\hat{\mathbb{F}}\) are IVCFSs and \(\hat{\mathbb{R}}\) denotes the IVCFR._  
**Example 5. **_The Cartesian product of two IVCFSs_  
\[\hat{\mathbb{E}}=\left\{ \begin{array}{c}\Big{(}s,[0.28,0.36]e^{i2[0.43,0.61] \pi}\Big{)},\Big{(}t,[0.17,0.23]e^{i2[0.67,0.81]\pi}\Big{)},\\ \Big{(}u,[0.45,0.50]e^{i2[0.04,0.21]\pi}\Big{)},\Big{(}v,[0.89,0.98]e^{i2[0.0,16]\pi}\Big{)}\end{array} \right\}\text{and}\]  
\[\hat{\mathbb{F}}=\left\{ \begin{array}{c}\Big{(}w,[0.53,0.64]e^{i2[0.29,0.3 7]\pi}\Big{)},\Big{(}x,[0.73,0.79]e^{i2[0.43,0.52]\pi}\Big{)},\\ \Big{(}y,[0.19,0.35]e^{i2[0.24,0.35]\pi}\Big{)},\Big{(}z,[0.39,0.55]e^{i2[0.45,0. 55]\pi}\Big{)}\end{array} \right\}\text{is}\]  
\[\hat{\mathbb{E}}\times\hat{\mathbb{F}}=\left\{ \begin{array}{c}\Big{(}(s,w),[0.28,0.36]e^{i2[0.29,0.37]\pi}\Big{)},\Big{(}(s,x),[0.28,0.36]e^{i2[0.43,0.52]\pi} \Big{)},\\ \Big{(}(s,y),[0.19,0.35]e^{i2[0.24,0.36]\pi}\Big{)},\Big{(}(s,z),[0.28,0.36]e^{i 2[0.016]\pi}\Big{)},\\ \Big{(}(t,w),[0.17,0.23]e^{i2[0.29,0.37]\pi}\Big{)},\Big{(}(t,x),[0.17,0.23]e^{i2 [0.43,0.52]\pi}\Big{)},\\ \Big{(}(t,y),[0.17,0.23]e^{i2[0.24,0.35]\pi}\Big{)},\Big{(}(t,z),[0.17,0.23]e^{i 2[0.45,0.55]\pi}\Big{)},\\ \Big{(}(u,w),[0.45,0.50]e^{i2[0.04,0.21]\pi}\Big{)},\Big{(}(u,x),[0.45,0.50]e^{i2[0.04,0.21]\pi}\Big{)},\\ \Big{(}(u,y),[0.19,0.35]e^{i2[0.04,0.21]\pi}\Big{)},\Big{(}(u,z),[0.39,0.50]e^{i2[0.04,0.21]\pi}\Big{)},\\ \Big{(}(v,w),[0.53,0.64]e^{i2[0.29,0.37]\pi}\Big{)},\Big{(}(v,x),[0.73,0.79]e^{i2[0.016]\pi}\Big{)},\\ \Big{(}(v,y),[0.19,0.35]e^{i2[0.016]\pi}\Big{)},\Big{(}(v,z),[0.39,0.55]e^{i2[0.016]\pi}\Big{)}\end{array} \right\}\]The IVCFR \(\bar{\mathbb{R}}\). between the IVCFSs \(\bar{\mathbb{E}}\) and \(\bar{\mathbb{F}}\) is given  
\[\bar{\mathbb{R}}=\left\{ \begin{array}{c}\left((s,w),[0.28,0.36]e^{i2[0.29,0.37] \pi}\right),\left((s,x),[0.28,0.36]e^{i2[0.43,0.52]\pi}\right),\\ \left((t,x),[0.17,0.23]e^{i2[0.43,0.52]\pi}\right),\left((u,w),[0.45,0.50]e^{i2 [0.04,0.21]\pi}\right),\\ \left((v,y),[0.19,0.35]e^{i2[0.16]\pi}\right),\left((v,z),[0.39,0.55]e^{i2[0.0 16]\pi}\right)\end{array} \right\}.\]  
**Definition 7**. : _Ref.