Given a binary relation $R$ over sets $A$ and $B$,then prove:
The complement of the converse relation is the converse of the complement relation,e.g: $$\overline{R^T}=\overline{R}^T$$
$R$ is defined as :$$R:=\left\{\left(a,b\right)\mid aRb\right\}$$
Then: $$R^T=\left\{\left(b,a\right)\mid aRb\right\}$$ $$\overline{R^T}=\left\{\left(b,a\right)\mid a\not Rb\right\}\tag{1}$$
On the other hand: $$\overline{R}=\left\{\left(a,b\right)\mid a\not Rb\right\}$$
$$\overline{R}^T=\left\{\left(b,a\right)\mid a\not Rb\right\}\tag{2}$$
Since $(\text{1})=(\text{2})$,implies the claim does hold.
Is the process true?
This is fine.
Note that the proof would normally be written similar to what follows (where $a\in A, b\in B$):
$(a,b)\in \overline{R^T} \iff (a,b)\not\in R^T \iff (b,a) \not \in R \iff (b,a)\in \overline{R} \iff (a,b) \in \overline{R}^T.$
Hence $\overline{R^T}=\overline{R}^T$.