(a) Given $A, B \subseteq \mathbb{N}$, define a many-one reduction of $A$ to $B$. Show that if $B$ is recursively enumerable (r.e.) and $A \leqslant_{m} B$ then $A$ is also recursively enumerable.

(b) State the $s-m-n$ theorem, and use it to prove that a set $X \subseteq \mathbb{N}$ is r.e. if and only if $X \leqslant_{m} \mathbb{K}$.

(c) Consider the sets of integers $P, Q \subseteq \mathbb{N}$ defined via

$$\begin{aligned} &P:=\left\{n \in \mathbb{N} \mid n \text { codes a program and } W_{n} \text { is finite }\right\} \\ &Q:=\left\{n \in \mathbb{N} \mid n \text { codes a program and } W_{n} \text { is recursive }\right\} \end{aligned}$$

Show that $P \leqslant \leqslant_{m} Q$.