If $R^*(R+S)\equiv S^*(R+S)$, then $R^*\equiv S^*$
I have to prove whether this is true or false.
Right now I think it's true, because I can't really think of a counter example. This is where I got:
$R^*(R+S)\equiv S^*(R+S)$, then $R^*\equiv S^*$ means
$$L(R^*)L(R+S)= L(S^*)L(R+S)$$
Where $(*)$ is the Kleene star and $L(R)$ means language denoted by $R$.
Don't really know if we can "cross out" on languages, since the $L(R+S)$ in common on both sides, our expression becomes
$$L(R^*)= L(S^*)$$
which means $R^*\equiv S^*$
would this be correct? The only step I am unsure about is when I "cross out" $L(R+S)$ since it is common on both sides.
Suppose $R$ is $\{\epsilon\}$ and $S$ is $\{a\}^*$.
Now $R^* = R$ and $S^* = SS = S$.
Also $(R+T) = RT = T$ for any $T$.
So $R^*(R+S) = S^*(R+S) = S$, but $R^* \ne S^*$