Can $L$ be regular language if it is a union of infinitely many regular languages $L_1,L_2,L_3,…$ over the same alphabet ?
(a) can $L$ be regular ?
(b) Is $L$ always regular ?
I want to make sure my logic is right. I am saying that the answer to both question is wrong because we will need construct FA $M$ that recognizes the union of the languages but since we have infinite number of states we can't construct M with infinite number of states.
Best Answer
$L$ certainly can be regular. Let $\Sigma$ be any finite alphabet, and let $L=\Sigma^*$; $L$ is countably infinite, so we can enumerate it as $L=\{w_n:n\in\Bbb Z^+\}$. For $n\in\Bbb Z^+$ let $L_n=\{w_n\}$; the language $L_n$ is finite, so it’s certainly regular. The language $\Sigma^*$ is also regular, and clearly $\Sigma^*=\bigcup_{n\in\Bbb Z^+}L_n$.
On the other hand, it need not be: the same reasoning shows that every infinite language is a union of infinitely many regular languages, and there are certainly infinite languages that are not regular.