Proving a set is not recursively enumerable

Prove that the set $\mathrm{TOTAL}$ is not recursively enumerable. Hint. For the universal Turing machine $\mathfrak {U}$, consider the function $\mathfrak {U}(f(x), x) \& a$ (concatenation with a nonempty symbol).

Assume that $A \subseteq \mathbf{N}$ is a set with the following properties: (i) $\phi_{n}$ is recursive for all $n \in A$, and (ii) for all recursive functions $f, f=\phi_{n}$ for some $n \in A$. Show that $A$ is not an r.e. set.

Show that the set $\left\{ \langle i, j\rangle \mid W_{i}=\overline{W_{j}}\right\}$ is not an r.e. set.

Use the recursion theorem to prove that the following sets are not r.e.: Fin, Rec, $\overline{\mathrm{REC}}$.

Prove that there exists a set that is not recursively enumerable.

Let $J=\left\{ w \mid \text{either } w=0 x \text{ for some } x \in A_{\text{TM }}, \text{ or } w=1 y \text{ for some } y \in \overline{A_{\text{TM }}}\right\}$. Show that neither $J$ nor $\bar{J}$ is Turing-recognizable.

Show that any infinite subset of $\operatorname {MIN}_{\mathrm{TM}}$ is not Turing-recognizable.

Show that the set of incompressible strings contains no infinite subset that is Turing-recognizable.

Let $S=\left\{ \langle M\rangle \mid M\text{ is a TM and }L(M)=\left\{ \langle M\rangle \right\} \right\}$. Show that neither $S$ nor $\bar{S}$ is Turing-recognizable.