# Is the Kleene star of an intersection always equal to the intersection of kleene stars?

I know that the Kleene star of an intersection is contained in the intersection of Kleene stars, but are they necessarily equal?

For example, given two formal languages, $$A$$ and $$B$$, I know that $$(A\cap B)^*\subseteq A^*\cap B^*$$, but does $$(A\cap B)^*= A^*\cap B^*$$?

• I know it is contained but is it equal =? – drugsrbad Sep 26 '19 at 21:04

Clearly not.

Let $$A=\{a\}$$ and $$B=\{aa\}$$.

Now,

$$A\cap B = \emptyset$$

so

$$(A\cap B)^* = \{\epsilon\}$$

but

$$A^*\cap B^*=B^*=\{a^{2i} : i \in \mathbb{N}\}$$

(all strings consisting of an even number of $$a$$).

• @ttnick: Sorry, I should have gone for "accept and improve". Thanks for bringing this to my attention. – rici Sep 30 '19 at 7:29