Your question makes little sense to someone unfamiliar with how Turing machines are coded as numbers in your course. You probably realize that there...
For each $b\in B$ choose a single $a_b\in A$ such that $f(a_b)=b$. Then consider the restriction of $f$ to $A'=\{a_b\mid b\in B\}$. Show that this is...
The book "Basic Set Theory" by A. Shen and N. K. Vereshchagin and this question from StackExchange say that the axiom of choice is needed to select a...