Let a relation $R$ be defined by $R=\left\{(4,5),(1,4),(4,6),(7,6),(3,7)\right\}$ then $R^{-1}oR$ is:

The correct option is A

$\left\{(1,1),(4,4),(4,7),(7,4),(7,7),(3,3)\right\}$

Explanation For The Correct Option:

Determining the composition relation of $R$ and $R^{-1}$

The given relation $R=\left\{(4,5),(1,4),(4,6),(7,6),(3,7)\right\}$

Considering this relation is defined on a set $A$.

We know that $R^{-1}=\left\{\left(y,x\right):(x,y)\in R\right\}$

Therefore,$R^{-1}=\left\{(5,4),(4,1),(6,4),(6,7),(7,3)\right\}$

We also know that composition of two $R$ and $S$ relation defined on a set $A$

$S\circ R=\left\{(x,y):\exists zsuchthat\in A(x,z)\in R\&(z,y)\in S\right\}$

Therefore,

$$\begin{gathered} \because (4,5)\in R\&(5,4)\in S\Rightarrow (4,4)\in R^{-1}\circ R \\ \because (1,4)\in R\&(4,1)\in S\Rightarrow (1,1)\in R^{-1}\circ R \\ \because (4,6)\in R\&(6,4)\in S\Rightarrow (4,4)\in R^{-1}\circ R \\ \because (7,6)\in R\&(6,7)\in S\Rightarrow (7,7)\in R^{-1}\circ R \\ \because (3,7)\in R\&(7,3)\in S\Rightarrow (3,3)\in R^{-1}\circ R \\ \because (4,6)\in R\&(6,7)\in S\Rightarrow (4,7)\in R^{-1}\circ R \\ \because (7,6)\in R\&(6,4)\in S\Rightarrow (7,4)\in R^{-1}\circ R \end{gathered}$$

Thus, $R^{-1}\circ R=\left\{(1,1),(4,4),(4,7),(7,4),(7,7),(3,3)\right\}$

Hence, option A is the correct answer.