Why is it true that the relation R and its negation are not semi decidable?
If the relation $R$ was semi-decidable, then the halting problem would be decidable: if you want to know whether program&nbsp;$x$ terminates, just enumerate the elements of&nbsp;$R$ until you find either $(x,0)$ or $(x,1)$. In the first case, output "$x$ halts"; in the second, "$x$ does not halt." One of these is guaranteed to happen, so you have a decider for the halting problem, contradicting the undecidability of that problem.

The argument for the complement of $R$ is nearly identical: observe that $\overline{R} = \{(x,1-y)\mid (x,y)\in R\}$.