Answer

Correct Option: (ii) sometimes (i.e., not always — CFLs are not closed under difference)

Explanation

$L_1 - L_2 = L_1 \cap \overline{L_2}$.

• CFLs are closed under union, concatenation, Kleene star, but NOT under intersection, complement, difference.
• Since complement of a CFL is not necessarily CFL, difference need not be CFL.

Counterexample:
$L_1 = a^*b^*c^*$ (actually regular, also CFL)
Let $L_1 = \{a^n b^n c^m\}$ (CFL) and $L_2 = \{a^m b^n c^n\}$ (CFL)
$L_1 \cap L_2 = \{a^n b^n c^n\}$ not CFL.
Then $L_1 - ( \Sigma^* - L_2)$ style constructions show difference can be non-CFL.

But sometimes it is CFL: e.g., if $L_2$ is regular, then $L_1 - L_2 = L_1 \cap \overline{L_2}$ is CFL because CFL ∩ Regular = CFL. So difference is sometimes CFL.

Many textbooks phrase options as:

• (i) always — false
• (ii) sometimes / not always closed
• (iii) never — false (counterexample above shows sometimes)

Hence correct is not necessarily CFL → Option (ii) sometimes (or “not always”). If exam key uses “never”, they mean “not closed”, also acceptable as (iii) in that phrasing — but strictly “sometimes” is precise.

Final Answer

$L_1 - L_2$ is not guaranteed CFL → Not closed → Option (ii) sometimes / not always (exam may list as (iii) never in weaker phrasing).