The language substr$($L$)$ is the set $\{$x : $$y in L for which x is a substring of y$\}.$ In each

case, show that your answer is correct.

$($a$)$ If L is Turing decidable, must substr$($L$)$ be Turing decidable?

$($b$)$ If L is Turing recognizable, must substr$($L$)$ be Turing recognizable?

$($c$)$ If substr$($L$)$ is Turing decidable, must L be Turing decidable?

$($d$)$ If substr$($L$)$ is Turing recognizable, must L be Turing recognizable?