Problem $7$

$($a$)$ Give an example of a context$-$free language whose complement is not context$-$free. Give a precise definition of this language and explain your answer. If such a language does not exist, explain why.

$($b$)$ Cive an example of a regular language that has a subset which is not context$-$free. Give a precise definition of both languages and explain your answer. If such a language does not exist, explain why.

$($c$)$ Give an example of a recursive language whose complement is not recursively enumerable. Give a precise definition of this language and explain your answer. If such a language does not exist, explain why.

$($d$)$ Give an example of a recursively enumerable language whose complement is not recursively enumer$-$able. Give a precise definition of this language and explain your answer. If such a language does not exist, explain why.