# formal languages – If \$A in mathrm{RE}\$ and \$A leq_m overline{A}\$ then \$Ain mathrm{R}\$

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:

1. Why does $$A leq_m overline{A}$$ imply $$overline{A} leq_m A$$?
2. I understand the following step (why $$overline{A}$$ is in RE).
3. Which theorem is used to deduce that $$A$$ is in R? (I’m not a student in this class)