$(22)$ For a language L$,$ let perm$($L$)$ be the language:

$\{$w : x in L$,|$w$|=|$x$|,$ and w is a permutation of the letters in x$\}.$

$($a$)$ If L is a Turing$-$decidable language, must perm$($L$)$ be Turing$-$decidable? Explain.

$($b$)$ If L is a Turing$-$recognizable language, must perm$($L$)$ be Turing$-$recognizable? Explain.

Hint: Use an enumerator for L$.$

$($c$)$ Is there a language L for which perm$($L$)$ is not Turing$-$recognizable? Explain. Hint.

Question $51$ might help.

$($d$)$ If perm$($L$)$ is a regular language, must L be Turing$-$recognizable? Explain.

$(51)$ A language L $\backslash$Sigma $?$ is unary if $|\backslash$Sigma $|=1.$ Show that there is a unary language that is not

Turing recognizable.