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Given a formal language L, is $L \subset L^*$ or is $L \subseteq L^*$?

To give context, I am tasked with proving whether or not there exists a language such that $(L^*)^c = (L^c)^*$. Assuming the logic behind my proof is correct, I've concluded that if $L \neq L^*$ then $(L^*)^c$ cannot equal $(L^c)^*$. I won't go into my proof as it's the not primary subject of my question (that is, unless someone is interested enough to check my work).

Thanks for any assistance!

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Does $\neg L$ stand for the complement of a language? I have seen $\bar L$, $L'$ and $L^c$ to denote complements, not $\neg L$. [I prefer the notation $L^c$ to the other two.] –  Srivatsan Dec 4 '11 at 0:18
    
Yes, it stands for the complement. I will edit the post to use a more familiar notation. –  bronxbomber92 Dec 4 '11 at 0:25
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1 Answer

up vote 2 down vote accepted

Both are possible. For example, we have $\{a\}\subsetneq \{a\}^*$, but $\{a^n\mid n\ge 0\}=\{a^n\mid n\ge 0\}^*$.

For your "contextual" question, consider $L=\varnothing$...

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Thanks! I had been considering when L is the null set, but I wasn't sure whether the null set was a valid language. –  bronxbomber92 Dec 4 '11 at 0:42
    
Why shouldn't it? –  Raphael Dec 4 '11 at 11:15
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