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 $(A\cap B)^*= A^*\cap B^*$?

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

Clearly not.

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


$A\cap B = \emptyset$


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


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

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

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

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