Match List-I with List-II:

List-I	List-II
(a) $\overline{L_3}\cup L_4$	(i) Context-free language
(b) $\overline{L_2}\cup L_3$	(ii) Recursively enumerable language
(c) $L_1^*\cap L_2$	(iii) Recursive language

where $L_1$: Regular language $L_2$: Context-free language $L_3$: Recursive language $L_4$: Recursively enumerable language Choose the correct option from those given below:

For $(a)$:

$L_3$ is a recursive language.

Complement of a recursive language is also recursive.

So,

$\overline{L_3}$ is recursive.

Also, $L_4$ is recursively enumerable.

Union of a recursive language and a recursively enumerable language is recursively enumerable.

Therefore,

$\overline{L_3}\cup L_4$ is recursively enumerable.

So,

$(a)\rightarrow(ii)$

For $(b)$:

$L_2$ is context-free language.

Every context-free language is recursive.

So, complement of $L_2$ is also recursive.

Also, $L_3$ is recursive.

Union of two recursive languages is recursive.

Therefore,

$\overline{L_2}\cup L_3$ is recursive.

So,

$(b)\rightarrow(iii)$

For $(c)$:

$L_1$ is regular.

Regular languages are closed under Kleene star.

So,

$L_1^*$ is regular.

Intersection of a regular language and a context-free language is context-free.

Therefore,

$L_1^*\cap L_2$ is context-free.

So,

$(c)\rightarrow(i)$

Hence, correct matching is

$(a)-(ii);\ (b)-(iii);\ (c)-(i)$