I have the following exercise:
Prove that $\{ab,aba\}^*=\{\epsilon\} \cup \{a\}\{ba,baa\}^*\{b,ba\}$.
My idea was to write the words of each of the languages in the following way,for example for the left side:
$(ab)^{n_0}(aba)(ab)^{n_1}(aba)...(aba)(ab)^{n_k}$
And then I try to make assumptions to $n_i$ and $k$. I think this is leading somewhere in the but isn't really satisfying. Can someone give me a hint on how to approach such proofs in a better way?
