Does $L_1L_2 = L_2L_1$ imply $L_1 = L_2$? Let $L_1, L_2 \subseteq \Sigma^*$ be two languages, where $\Sigma$ is some finite Alphabet.

Does $L_1L_2 = L_2L_1$ imply $L_1 = L_2$?

What if $L_1$ and $L_2$ are regular languages?

Can you give counterexamples?
No, just take as counter example $L_1=\epsilon$ and $L_2$ any other langage different from $L_1$.