## 3 Interval-Valued Complex Fuzzy Relations
\begin{array}{l}x,\left[\alpha_{\text{(\hat{E} )\text{m}}}^{-}(x),\alpha_{\text{(\hat{E})\text{m}}  
\[\hat{\mathbb{E}}\times\hat{\mathbb{F}}=\left\{\begin{array}{l}\left((u,x),[0.07,0.12 ]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.63,0.74]\pi}\right),\\ \left((u,y),[0.07,0.12]e^{i2[0.24,0.29]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((u,z),[0.07,0.12]e^{i2[0.24,0.29]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((v,x),[0.26,0.34]e^{i2[0.03,0.15]\pi},[0.38,0.49]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,y),[0.52,0.58]e^{i2[0.03,0.15]\pi},[0.18,0.37]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,z),[0.28,0.36]e^{i2[0.03,0.15]\pi},[0.48,0.61]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,x),[0.26,0.34]e^{i2[0.15,0.25]\pi},[0.38,0.49]e^{i2[0.63,0.74]\pi} \right),\\ \left((w,y),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right),\\ \left((w,z),[0.28,0.36]e^{i2[0.15,0.25]\pi},[0.48,0.61]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\right.\]  
The IVCIFR \(\hat{\mathbb{R}}\) between the IVCFS \(\hat{\mathbb{E}}\) and \(\hat{\mathbb{F}}\) is  
\[\hat{\mathbb{R}}=\left\{\begin{array}{l}\left((u,y),[0.07,0.12 ]e^{i2[0.24,0.29]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi}\right),\\ \left((v,x),[0.26,0.34]e^{i2[0.03,0.15]\pi},[0.38,0.49]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,z),[0.28,0.36]e^{i2[0.03,0.15]\pi},[0.48,0.61]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,y),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right),\\ \left((w,z),[0.28,0.36]e^{i2[0.15,0.25]\pi},[0.48,0.61]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\]  
The IVCIFR \(\hat{\mathbb{R}}\) between the IVCFS \(\hat{\mathbb{E}}\) and \(\hat{\mathbb{F}}\) is given as below (Figure 1),  
**NOTE:** For convenience, throughout this article, \(x\) and \((x,y)\) will be used to denote \(\left(x,\left[\alpha^{-}_{(\hat{\mathbb{E}})\hat{\mathbb{m}}}(x),\alpha^{+}_{( \hat{\mathbb{E}})\hat{\mathbb{m}}}(x)\right]\right)\) and \(\left(\begin{array}{c}(x,y),\left[\alpha^{-}_{(\hat{\mathbb{E}}\times \hat{\mathbb{E}})\hat{\mathbb{m}}}(x,y),\alpha^{+}_{(\hat{\mathbb{E}}\times \hat{\mathbb{E}})\hat{\mathbb{m}}}(x,y)\right]e^{i2[0.\hat{\mathbb{E}}\cdot \hat{\mathbb{E}},\hat{\mathbb{E}}\cdot(x,y),\alpha^{+}_{(\hat{\mathbb{E}}\cdot \hat{\mathbb{E}})\hat{\mathbb{m}}}(x,y)]\pi},\\ \left[\alpha^{-}_{(\hat{\mathbb{E}}\times\hat{\mathbb{E}})\hat{\mathbb{m}}}(x,y),\alpha^{+}_{(\hat{\mathbb{E}}\times\hat{\mathbb{E}})\hat{\mathbb{m}}}(x,y)\right]e^{i2[0.\hat{\mathbb{E}}\cdot\hat{\mathbb{E}},\hat{\mathbb{E}}\cdot \hat{\mathbb{E}}\cdot(x,y),\alpha^{+}_{(\hat{\mathbb{E}}\cdot\hat{\mathbb{E}}) \hat{\mathbb{m}}}(x,y)]\pi}\end{array}\right)\), respectively, otherwise it will be mentioned.  
Figure 1: Interval-valued complex intuitionistic fuzzy relation.  
**Definition 12**.: _Let \(\tilde{\mathbb{E}}\) be an IVCIFS in a universe \(T\) and \(\tilde{\mathbb{R}}\) be an IVCIFR on \(\tilde{\mathbb{E}}\). Then,_  
1. \(\tilde{\mathbb{R}}\) _is known as an IVCI reflexive fuzzy relation (IVCI-reflexive-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \((x,x)\in\tilde{\mathbb{R}}\)_,_ \(\forall x\in\tilde{\mathbb{E}}\)_._
2. \(\tilde{\mathbb{R}}\) _is known as an IVCI irreflexive fuzzy relation (IVCI-irreflexive-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \((x,x)\
otin\tilde{\mathbb{R}}\)_,_ \(\forall x\in\tilde{\mathbb{E}}\)_._
3. \(\tilde{\mathbb{R}}\) _is known as an IVCI symmetric fuzzy relation (IVCI-symmetric-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\forall x,y\in\tilde{\mathbb{E}}\)_,_ \((x,y)\in\tilde{\mathbb{R}}\Rightarrow(y,x)\in\tilde{\mathbb{R}}\)_._
4. \(\tilde{\mathbb{R}}\) _is known as an IVCI antisymmetric fuzzy relation (IVCI-antisymmetric-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\forall x,y\in\tilde{\mathbb{E}}\)_,_ \((x,y)\in\tilde{\mathbb{R}}\) _and_ \((y,x)\in\tilde{\mathbb{R}}\Rightarrow x=y\)_._
5. \(\tilde{\mathbb{R}}\) _is known as an IVCI asymmetric fuzzy relation (IVCI-asymmetric-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\forall x,y\in\tilde{\mathbb{E}}\)_,_ \((y,x)\in\tilde{\mathbb{R}}\Rightarrow(x=y\
otin\tilde{\mathbb{R}})\)_._
6. \(\tilde{\mathbb{R}}\) _is known as an IVCI complete fuzzy relation (IVCI-complete-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\forall x,y\in\tilde{\mathbb{R}}\)_,_ \((x,y)\in\tilde{\mathbb{R}}\) _or_ \((y,x)\in\tilde{\mathbb{R}}\)_._
7. \(\tilde{\mathbb{R}}\) _is known as an IVCI transitive fuzzy relation on (IVCI-transitive-FR)_ \(\tilde{\mathbb{E}}\) _if_ \(\forall x,y,z\in\tilde{\mathbb{E}}\)_,_ \((x,y)\in\tilde{\mathbb{R}}\) _and_ \((y,z)\in\tilde{\mathbb{R}}\Rightarrow(x,z)\in\tilde{\mathbb{R}}\)_._
8. \(\tilde{\mathbb{R}}\) _is known as an IVCI equivalence fuzzy relation (IVCI-equivalence-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\tilde{\mathbb{R}}\) _is IVCI-reflexive-FR, IVCI-symmetric-FR and IVCI-transitive-FR on_ \(\tilde{\mathbb{E}}\)_._
9. \(\tilde{\mathbb{R}}\) _is known as an IVCI preorder fuzzy relation (IVCI-preorder-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\tilde{\mathbb{R}}\) _is IVCI-reflexive-FR and IVCI-transitive-FR on_ \(\tilde{\mathbb{E}}\)_._
10. \(\tilde{\mathbb{R}}\) _is known as an IVCI strict order fuzzy relation (IVCI-strict order-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\tilde{\mathbb{R}}\) _is IVCI-irreflexive-FR and IVCI-transitive-FR on_ \(\tilde{\mathbb{E}}\)_._
11. \(\tilde{\mathbb{R}}\) _is known as an IVCI partial order fuzzy relation (IVCI-partial order-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\tilde{\mathbb{R}}\) _is IVCI-preorder-FR and IVCI-antisymmetric-FR on_ \(\tilde{\mathbb{E}}\)_._
12. \(\tilde{\mathbb{R}}\) _is known as an IVCI linear order fuzzy relation (IVCI-linear order-FR) on_ \(\tilde{\mathbb{E}}\) _if_ \(\tilde{\mathbb{R}}\) _is IVCI-partial order-FR and IVCI-complete-FR on_ \(\tilde{\mathbb{E}}\)_._  
**Example 10**.: _For an IVCIFS \(\tilde{\mathbb{E}}=\left\{\begin{array}{l}\left(u,[0.07,0.12]e^{i2[0.15,0.2 5]\pi},[0.56,0.66]e^{i2[0.63,0.74]\pi}\right),\\ \left(v,[0.07,0.12]e^{i2[0.03,0.15]\pi},[0.56,0.66]e^{i2[0.73,0.82]\pi}\right),\\ \left(u,w,[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left(v,[v,u],[0.07,0.12]e^{i2[0.03,0.15]\pi},[0.56,0.66]e^{i2[0.57,0.82]\pi} \right),\\ \left((v,v),[0.07,0.12]e^{i2[0.03,0.15]\pi},[0.56,0.66]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,v),[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,w),[0.37,0.46]e^{i2[0.03,0.15]\pi},[0.32,0.43]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,u),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((w,v),[0.37,0.46]e^{i2[0.03,0.15]\pi},[0.32,0.43]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,w),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\)_  
Then,1. The IVCI-equivalence-FR \(\bar{\mathbb{R}}_{1}\) on \(\bar{\mathbb{E}}\) is given as \[\bar{\mathbb{R}}_{1}=\left\{\begin{array}{l}\left((u,u),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.63,0.74]\pi}\right),\\ \left((u,w),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((v,v),[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,u),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((w,w),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\]
2. The IVCI-preorder-FR \(\bar{\mathbb{R}}_{2}\) on \(\bar{\mathbb{E}}\) is given as \[\bar{\mathbb{R}}_{2}=\left\{\begin{array}{l}\left((u,u),[0.07,0.12]e^{i2[0.1 5,0.25]\pi},[0.56,0.66]e^{i2[0.63,0.74]\pi}\right),\\ \left((v,u),[0.07,0.12]e^{i2[0.03,0.15]\pi},[0.56,0.66]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,v),[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,w),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\]
3. The IVCI-strict order-FR \(\bar{\mathbb{R}}_{3}\) on \(\bar{\mathbb{E}}\) is given as \[\bar{\mathbb{R}}_{3}=\left\{\begin{array}{l}\left((v,u),[0.07,0.12]e^{i2[0. 03,0.15]\pi},[0.56,0.66]e^{i2[0.73,0.82]\pi}\right),\\ \left((w,u),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((w,v),[0.37,0.46]e^{i2[0.03,0.15]\pi},[0.32,0.43]e^{i2[0.73,0.82]\pi} \right)\end{array}\right\}\]
4. The IVCI-partial order-FR \(\bar{\mathbb{R}}_{4}\) on \(\bar{\mathbb{E}}\) is given as \[\bar{\mathbb{R}}_{4}=\left\{\begin{array}{l}\left((u,u),[0.07,0.12]e^{i2[0.1 5,0.25]\pi},[0.56,0.66]e^{i2[0.63,0.74]\pi}\right),\\ \left((v,u),[0.07,0.12]e^{i2[0.03,0.15]\pi},[0.56,0.66]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,v),[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,u),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((w,w),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\]
5. The IVCI-linear order-FR \(\bar{\mathbb{R}}_{5}\) on \(\bar{\mathbb{E}}\) is given as \[\bar{\mathbb{R}}_{5}=\left\{\begin{array}{l}\left((u,u),[0.07,0.12]e^{i2[0.1 5,0.25]\pi},[0.56,0.66]e^{i2[0.63,0.74]\pi}\right),\\ \left((v,u),[0.07,0.12]e^{i2[0.03,0.15]\pi},[0.56,0.66]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,v),[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,u),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((w,v),[0.37,0.46]e^{i2[0.03,0.15]\pi},[0.32,0.43]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,w),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\]  
**Definition 13**.: _For an IVCIFR \(\bar{\mathbb{R}}\), the converse relation \(\bar{\mathbb{R}}^{c}\) of \(\bar{\mathbb{R}}\) is defined as, \(\bar{\mathbb{R}}^{c}=\left\{\left(y,x\right):(x,y)\in\bar{\mathbb{R}}\right\}\)._  
**Example 11**.: _If \(\mathbb{R}=\left\{\begin{array}{l}\left((u,y),[0.07,0.12]e^{i2[0.24,0.29]\pi}, [0.56,0.66]e^{i2[0.57,0.71]\pi}\right),\\ \left((v,x),[0.26,0.34]e^{i2[0.03,0.15]\pi},[0.38,0.49]e^{i2[0.73,0.82]\pi} \right),\\ \left((v,z),[0.28,0.36]e^{i2[0.03,0.15]\pi},[0.48,0.61]e^{i2[0.73,0.82]\pi} \right),\\ \left((w,y),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right),\\ \left((w,z),[0.28,0.36]e^{i2[0.15,0.25]\pi},[0.48,0.61]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\) is an IVCIFR between IVCIFSs \(\mathbb{\hat{E}}=\left\{\begin{array}{l}\left(u,[0.07,0.12]e^{i2[0.24,0.29] \pi},[0.56,0.66]e^{i2[0.57,0.71]\pi}\right),\\ \left(v,[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left(w,[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\) and_  
\(\mathbb{\hat{F}}=\left\{\begin{array}{l}\left(x,[0.26,0.34]e^{i2[0.15,0.25] \pi},[0.38,0.49]e^{i2[0.63,0.74]\pi}\right),\\ \left(y,[0.52,0.58]e^{i2[0.30,0.42]\pi},[0.18,0.37]e^{i2[0.06,0.13]\pi}\right),\\ \left(z,[0.28,0.36]e^{i2[0.43,0.53]\pi},[0.48,0.61]e^{i2[0.32,0.42]\pi}\right),\end{array}\right\}\), then the converse relation \(\mathbb{R}^{c}\) of \(\mathbb{\hat{R}}\) is  
\(\mathbb{\hat{R}}^{c}=\left\{\begin{array}{l}\left((x,v),[0.26,0.34]e^{i2[0.03,0.15]\pi},[0.38,0.49]e^{i2[0.73,0.82]\pi}\right),\\ \left((y,u),[0.07,0.12]e^{i2[0.24,0.29]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((y,w),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right),\\ \left((z,v),[0.28,0.36]e^{i2[0.03,0.15]\pi},[0.48,0.61]e^{i2[0.73,0.82]\pi} \right),\\ \left((z,w),[0.28,0.36]e^{i2[0.15,0.25]\pi},[0.48,0.61]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\)  
The IVCI-equivalence-FRs sets off the concept of IVCIF-equivalences classes, which are defined as follows.  
**Definition 14**.: _For an IVCI-equivalence-FR \(\mathbb{\hat{R}}\), the IVCIF equivalence class of \(x\) modulo \(\mathbb{\hat{R}}\) is defined as, \(\mathbb{\hat{R}}[x]=\left\{y|(y,x)\in\mathbb{\hat{R}}\right\}\)._  
**Example 12**.: _If \(\mathbb{\hat{R}}=\left\{\begin{array}{l}\left((u,u),[0.07,0.12]e^{i2[0.15,0.2 5]\pi},[0.56,0.66]e^{i2[0.63,0.74]\pi}\right),\\ \left((u,w),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((v,v),[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left((u,u),[0.07,0.12]e^{i2[0.15,0.25]\pi},[0.56,0.66]e^{i2[0.57,0.71]\pi} \right),\\ \left((w,w),[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right),\end{array}\right\}\) is an IVCIFR on an IVCIFS \(\mathbb{\hat{E}}=\left\{\begin{array}{l}\left(u,[0.07,0.12]e^{i2[0.24,0.29] \pi},[0.56,0.66]e^{i2[0.57,0.71]\pi}\right),\\ \left(v,[0.63,0.71]e^{i2[0.03,0.15]\pi},[0.11,0.23]e^{i2[0.73,0.82]\pi} \right),\\ \left(w,[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\), then the IVCIF-equivalences class of \(1.\)\(x\) modulo \(\mathbb{\hat{R}}\) is given as  
\(\mathbb{\hat{R}}[u]=\left\{\begin{array}{l}\left(u,[0.07,0.12]e^{i2[0.24,0.29] \pi},[0.56,0.66]e^{i2[0.57,0.71]\pi}\right),\\ \left(w,[0.37,0.46]e^{i2[0.15,0.25]\pi},[0.32,0.43]e^{i2[0.53,0.68]\pi} \right)\end{array}\right\}\)
2. \(y\) modulo \(\mathbb{\hat{R}}\) is given as3. \(z\) _modulo_\(\bar{\mathbb{R}}\) is given as \[\bar{\mathbb{R}}[w]=\left\{\begin{array}{l}\left(u,[0.07,0.12]e^{2[0.24,0.29] \pi},[0.56,0.66]e^{2[0.57,0.71]\pi}\right),\\ \left(w,[0.37,0.46]e^{2[0.15,0.25]\pi},[0.32,0.43]e^{2[0.53,0.68]\pi}\right) \end{array}\right\}\]  
**Definition 15**.: _For an IVCIFR \(\bar{\mathbb{R}}\) on an IVCIFS \(\bar{\mathbb{E}}\), then the IVCI-composite-FR \(\bar{\mathbb{R}}\circ\bar{\mathbb{R}}\) is defined as: for each \((x,y)\in\bar{\mathbb{R}}\) and \((y,z)\in\bar{\mathbb{R}}\Rightarrow(x,z)\in\bar{\mathbb{R}}\circ\bar{\mathbb{R }},\forall x,y,z\in T\)._  
**Example 13**.: _For some IVCIFRs_  
\[\bar{\mathbb{R}}_{1}=\left\{\begin{array}{l}\left((u,v),[0.07,0.12]e^{2[0.03,0.15]\pi},[0.56,0.66]e^{2[0.73,0.82]\pi}\right),\\ \left((v,u),[0.07,0.12]e^{2[0.03,0.15]\pi},[0.56,0.66]e^{2[0.73,0.82]\pi}\right),\\ \left((w,v),[0.37,0.46]e^{2[0.03,0.15]\pi},[0.32,0.43]e^{2[0.73,0.82]\pi}\right) \end{array}\right\}\text{and}\\ \bar{\mathbb{R}}_{2}=\left\{\begin{array}{l}\left((u,w),[0.07,0.12]e^{2[0.15, 0.25]\pi},[0.56,0.66]e^{2[0.57,0.71]\pi}\right),\\ \left((v,u),[0.07,0.12]e^{2[0.03,0.15]\pi},[0.56,0.66]e^{2[0.73,0.82]\pi}\right),\\ \left((w,v),[0.37,0.46]e^{2[0.03,0.15]\pi},[0.32,0.43]e^{2[0.73,0.82]\pi}\right) \end{array}\right\}\text{on an IVCIFS}.\\ \left((w,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}