I'm trying to do the following demonstration, but I don't know where to start to formally demonstrate it:
Given $\Sigma=\lbrace a,b\rbrace ,L_1,L_2\subseteq \Sigma^*$ and
- $L_1 = \lbrace w\in\Sigma^* / |w|_a = 2 \rbrace$
- $L_2 = \lbrace w\in\Sigma^* / |w|_a = 2\cdot k ,k\in\mathbb{N} \rbrace$
Prove: $(L_1 \cup \lbrace b \rbrace)^* = L_2$
I get the idea that is true since Kleene's star will always take items from $ L_1 $ and this one always has an even amount of a's. Any idea how to prove it formally? :P