I've encountered a question during my studies: If $L\cdot\{\epsilon,0\}$ regular language, is $L$ regular?
I thought to disprove it by using $A\subseteq 2\mathbb{N}, L=\{w\in\{0\}^*:|w|\notin A\}$ but I need to prove that L is irregular and using the pumping lemma in my proof is forbidden. Any help?
If you’re not allowed to use basic tools like the pumping lemma and the Myhill-Nerode theorem, the only thing that I can suggest is a cardinality argument. There are only countably many regular expressions over the alphabet $\{0\}$, so there are only countably many regular languages that are subsets of $\{0\}^*$. However, there are uncountably many subsets $A\subseteq 2\Bbb N$, and each gives rise to a different language $L_A=\{w\in\{0\}^*:|w|\notin A\}$, so there must be an $A\subseteq 2\Bbb N$ such that $L_A$ is not regular.