$($Mapping Reductions$)$ Prove the following general statements about the existence of mapping

reductions.

$($a$)$ Let $A$ be an arbitrary recognizable language, and suppose $A\frac{?}{b}{?}_{m}ar(A).$ Show that $A$ is decidable.

$($Hint: Recall that a language is decidable iff both it and and its complement are recognizable.$)$

$($b$)$ Let $A$ be an arbitrary decidable language and let $B$ be an arbitrary language other than $\frac{O}{?}$ or

${}^{**}.($That is$,$ there exists a string $w_0!$inB and a string $w_1$inB.$)$ Show that $A{?}_{m}B.$