For any language $L,$ define the operation Double$(L)=\{w$@$w|winL\}$

$($a$)$ Show that the set of regular languages is not closed under the operation of Double.

$($Hint: find a regular language A such that Double$(A)$ is non$-$regular.$)$

$($b$)$ Show that the set of decidable languages is closed under the operation of Double.

$($Your work should include a construction but you do not have to do the proof of correctness part.

You should include an argument about why the construction is a decider.$)$

$($c$)$ Show that the set of recognizable languages is closed under the operation of Double.

$($Your work should include a construction but you do not have to do the proof of correctness part.$)$