Consider alphabet $=\{0,1\},$ and let $L$ be a language over $.$ Show that language $L$ is Turing recognizable if and only if there is a decidable language $L^{\text{'}}$ over ${\displaystyle \underset{?}{\overset{?}{}}}$ such that $L=\{\begin{array}{ccc}\text{w}\text{i}\text{n}& {}^{**}\text{:}(\text{:}u,& w\text{:})in{L}^{\text{'}}\end{array}$ for some string $\{$:uin${}^{**}\}.$

On the one hand, assume that $L^{\text{'}}$ satisfying the property in question exists. A TM recognizing $L$ operates on