I'm currently learning about automata theory and I came across the following question:
Given an arbitrary set $L$ of symbols. Is $(L \: \cdot \: L)$* = $L$* $\cdot$ $L$* true, where * refers to the Kleene star and $\cdot$ to concatenation?
I've already read in other questions, that $(L_1 \: \cdot \: L_2)$* = $L_1$* $\cdot$ $L_2$* is not true, however what happens when $L_1 = L_2$ as in this case. I feel like if its the same set then it should be true, but I'm not sure.
Hint. If $A$ is the alphabet, $(AA)^*$ is the set of words of even length. Now, what is $A^*A^*$?