## 3 Interval-Valued Complex Fuzzy Relations
: _Suppose \(\bar{\mathbb{R}}\) is an IVCI-partial order-FR on an IVCIFS \(\bar{\mathbb{E}}\), then the converse relation \(\bar{\mathbb{R}}^{c}\) of \(\bar{\mathbb{R}}\) is also an IVCI-partial order-FR on \(\bar{\mathbb{E}}\)._  
**Proof. **: In order to prove the assertion, it is sufficient to show that the converse of a complex intuitionistic partial order fuzzy relation \(R^{c}\) satisfies the three properties of a complex intuitionistic partial order fuzzy relation. By using the properties of IVCI-partial order-FR \(\bar{\mathbb{R}}\), we prove the statement. 1. It is given that \(\bar{\mathbb{R}}\) is an IVCI-reflexive-FR. Therefore, for any \(u\in\mathbb{S}\), \((u,u)\in\bar{\mathbb{R}}\Rightarrow(u,u)\in\bar{\mathbb{R}}^{c}\). Thus, \(\bar{\mathbb{R}}^{c}\) is an IVCI-reflexive-FR. 2. Assume that \((u,u)\in\bar{\mathbb{R}}^{c}\) and \((v,u)\in\bar{\mathbb{R}}^{c}\), then, \((u,v)\in\bar{\mathbb{R}}\) and \((v,u)\in\bar{\mathbb{R}}\). However, \(\bar{\mathbb{R}}\) is an IVCI-antisymmetric-FR. Therefore, \((u,v)=(v,u)\). Thus, \(\bar{\mathbb{R}}\) is also an IVCI-antisymmetric-FR. 3. Suppose that \((u,v)\in\bar{\mathbb{R}}^{c}\) and \((v,w)\in\bar{\mathbb{R}}^{c}\), then, \((w,v)\in\bar{\mathbb{R}}\) and \((v,u)\in\bar{\mathbb{R}}\). However, it is given that \(\bar{\mathbb{R}}\) is an IVCI-transitive-FR. Therefore, \((w,u)\in\bar{\mathbb{R}}\Rightarrow(u,w)\in\bar{\mathbb{R}}^{c}\). Thus, \(\bar{\mathbb{R}}^{c}\) is also an IVCI-transitive-FR. From i, ii and iii, the converse relation \(\bar{\mathbb{R}}^{c}\) of an IVCI-partial order-FR \(\bar{\mathbb{R}}\) is proved to be an IVCI-partial order-FR too. \(\square\)  
**Theorem 5**. : _Suppose \(\bar{\mathbb{R}}\) is an IVCI-equivalence-FR on an IVCIFS \(\bar{\mathbb{E}}\), then \((x,y)\in\bar{\mathbb{R}}\), iff \(\bar{\mathbb{R}}[u]=\bar{\mathbb{R}}[v]\)._  
**Proof. **: Assume that \((u,v)\in\bar{\mathbb{R}}\) and \(w\in\bar{\mathbb{R}}[u]\,\bar{\mathbb{R}}(w,u)\in\bar{\mathbb{R}}\). Now, by using the fact that an IVCI-equivalence-FR is also an IVCI-transitive-FR, so  
\((w,v)\in\bar{\mathbb{R}}\Rightarrow w\in\bar{\mathbb{R}}[v]\). Thus,  
\[\bar{\mathbb{R}}[u]\subseteq R[v] \tag{3}\]  
As \((u,v)\in\bar{\mathbb{R}}\), by using the fact that an IVCI-equivalence-FR is also an IVCI-symmetric-FR, so  
\((v,u)\in\bar{\mathbb{R}}\). Additionally, assume that \(w\in\bar{\mathbb{R}}[v]\Rightarrow(w,v)\in\bar{\mathbb{R}}\). Now, again by using the fact that an IVCI-equivalence-FR is also an IVCI-transitive-FR, so  
\((w,u)\in\bar{\mathbb{R}}\Rightarrow w\in\bar{\mathbb{R}}[u]\). Thus,  
\[\bar{\mathbb{R}}[v]\subseteq\bar{\mathbb{R}}[u] \tag{4}\]  
Therefore, (3) and (4) infer that \(\bar{\mathbb{R}}[v]=\bar{\mathbb{R}}[u]\). Conversely, assume that \(\bar{\mathbb{R}}[v]=\bar{\mathbb{R}}[u]\), \(w\in\bar{\mathbb{R}}[u]\) and \(w\in\bar{\mathbb{R}}[v]\Rightarrow(w,v)\in\bar{\mathbb{R}}\) and \((w,u)\in\bar{\mathbb{R}}\). Again, by using the fact that an IVCI-equivalence-FR is also an IVCI-symmetric-FR, so \((w,u)\in\bar{\mathbb{R}}\Rightarrow(u,w)\in\bar{\mathbb{R}}\). Now, by the definition of IVCI-transitive-FR,  
\((u,w)\in\bar{\mathbb{R}}\) and \((w,v)\in\bar{\mathbb{R}}\Rightarrow(u,v)\in\bar{\mathbb{R}}\),  
which completes the proof. \(\square\)