Disprove that $L_1^*\cup L_2^* = (L_1\cup L_2)^*$.

Question

Disprove that $L_1^*\cup L_2^* = (L_1\cup L_2)^*$.

i.e to find counter example.

I started with $L_1=\{a\}, L_2=\{b\}$ but it didn't worked.

Answer 1

Your example does work: $ab \in \{a, b\}^*$ but $ab \not\in \{a\}^*\cup \{b\}^*$.

Answer 2

There is even a counterexample on a one-letter alphabet. For instance, $(a^2)^* \cup (a^3)^* \not= (a^2, a^3)^*$ since $a^5 = a^2a^3 \in (a^2,a^3)^*$.