[Math] Is the set of regular languages closed under set difference

Question

Let $\Sigma$ be an alphabet and $L_1,L_2 \subseteq \Sigma^*$ two regular languages.

I know that $REG$ is closed under intersections of regular languages and under complementation of a regular language. My reasoning looks like this:

$L_1 \setminus L_2 = L_1 \cap \overline L_2$

Hence, we have that $REG$ is closed under set differentiation.

Have I overlooked something?

Answer 1

Your proof is correct and probably the easiest way to go about it.

Note that your argument is not specific to regular languages: if a class of subsets of a set is closed under intersection and complementation, then it is also closed under set difference (and under union)