I have a problem that says: Prove that exists only one language L where for every alphabet Σ, we have that L = complement(ΣL). (L= (ΣL)').
I know that the complement of a language L, is L' = Σ* - L, but with this definition i don't know how to find the language that satisfies the condition because Σ* is an infinite language and this confuses me. I also find out that the complement of a recursive language is the same recursive language but i don't know if this have any relation.
$L$ contains the empty word because $\Sigma L$ cannot. Then $\Sigma L$ contains each one-letter word. That means $L$ contains none of them. So, $\Sigma L$ contains no two-letters word. So $L$ contains them all. And so on...