Theory Of Computation - recognizable and decidable

Question

How to prove that for any language $A$, if $A$ is recognizable and $A \leq_m A^\complement$, then $A$ is decidable.

I know this theorem - A language is decidable iff both it and its complement are recognizable

How to explain this?

Answer 1

Let us recall some facts.

Definition 1: We write $A \leq_m B$ if there is a computable function $f$ such that for every $w$, $w \in A \iff f(w) \in B$.

Theorem 1: If $A \leq_m B$ and $B$ is recognizable, then $A$ is recognizable.

Theorem 2: A language $A$ is decidable iff $A$ and $\bar{A}$ is recognizable.

Now assume that $A$ is recognizable and $A \leq_m \bar{A}$. Notice that from Definition 1, $w \in A \iff f(w) \in \bar{A}$ also implies that $w \in \bar{A} \iff f(w) \in A$, so $\bar{A} \leq_m A$. Then, from Theorem 1, we can conclude that $\bar{A}$ is recognizable, and therefore we get from Theorem 2 that $A$ is decidable.