I found the following question with an answer here, but I can't understand the steps of the solution.
Show that if a language $A$ is in RE and $A \leq_m \overline{A}$, then $A$ is recursive.
Solution. Since $A \leq_m \overline{A}$, it follows that $\overline{A} \leq_m A$, and since $A$ is in RE, it follows that $\overline{A}$ is also in RE. Since both $A$ and $\overline{A}$ are in RE, it follows that $A$ is in R (this follows from a theorem you learned in class).
Here $\le_m$ demotes mapping reducibility.
Actually I can't understand most of the answer. In particular:
- Why does $A \leq_m \overline{A}$ imply $\overline{A} \leq_m A$?
- I understand the following step (why $\overline{A}$ is in RE).
- Which theorem is used to deduce that $A$ is in R? (I'm not a student in this class)