Let $\mathrm{L}$ be a language over $\Sigma$ and $c h_{\mathrm{L}}$
$$
c h_{\mathrm{L}}(w)= \begin{cases}1 & \text { if } w \in \mathrm{L} \\ 0 & \text { otherwise }\end{cases}
$$
be the characteristic function of $\mathrm{L}$.
a) Let $\mathrm{M}$ be a Turing machine that computes $c h_{\mathrm{L}}$. Prove that $\mathrm{L}$ is recursive.
b) If $\mathrm{L}$ is recursive, prove that there is a Turing machine that computes $c h_{\mathrm{L}}$.