[Solved]: Do Kleene star and complement commute?

Problem Detail: 

I am having hard time solving the following problem.

Are there any languages for which $$ \overline{L^*} = (\overline{L})^* $$

Assuming $\emptyset^* = \emptyset$, if I consider $\Sigma = \{a\}$ and L = $\Sigma^*$, I get that $L^* = L$ and that $\overline{L^*} = \emptyset$. For the right side I get $\overline{L} = \emptyset$ and $(\overline{L})^* = \emptyset$. Thus, both sides are equal.

Is it true that $\emptyset^* = \emptyset$?

Asked By : user118837

Answered By : Rick Decker

Hint: The star of a language always contains the empty string. The complement of a language containing the empty string never does. With that in mind, look at the left and the right hand sides of your proposed equality.

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Question Source : http://cs.stackexchange.com/questions/21544