(a) Define a recursive set and a recursively enumerable (r.e.) set. Prove that $E \subseteq \mathbb{N}_{0}$ is recursive if and only if both $E$ and $\mathbb{N}_{0} \backslash E$ are r.e. sets.

(b) Let $E=\left\{f_{n, k}\left(m_{1}, \ldots, m_{k}\right) \mid\left(m_{1}, \ldots, m_{k}\right) \in \mathbb{N}_{0}^{k}\right\}$ for some fixed $k \geqslant 1$ and some fixed register machine code $n$. Show that $E=\left\{m \in \mathbb{N}_{0} \mid f_{j, 1}(m) \downarrow\right\}$ for some fixed register machine code $j$. Hence show that $E$ is an r.e. set.

(c) Show that the function $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$ defined below is primitive recursive.

$$f(n)= \begin{cases}n-1 & \text { if } n>0 \\ 0 & \text { if } n=0\end{cases}$$

[Any use of Church's thesis in your answers should be explicitly stated. In this question $\mathbb{N}_{0}$ denotes the set of non-negative integers.]