what is the complement of empty language?

Question

If R- Regular language , C-Context Free language and L -Recursive language then what is the result of the expression ((R-C)-L)',Now first starting with R-C , It will give result as ∅, since every regular language is context free ,Now when I do ∅-L ,I get again ∅, Now finally when we calculate the complement of ∅, what should be the resulting nature of language , according to me it should be regular since complement of ∅, implies ∑- ∅ which should be ∑, so I guess it should be regular ,Is there anything wrong in this approach ?

Answer 1

"R- Regular language , C-Context Free language and L -Recursive language". Here they are specifying a SINGLE language of the respective type.Not the whole set.

If the Statement would be "R- Set of all Regular Languages, C-Set of all Context Free language and L -Set of all Recursive language", only then your reduction is CORRECT.

The answer for the question would be a Recursive language.

Answer 2

A language on the alphabet $\Sigma$ is a subset of $\Sigma^*$. Therefore, The complement of a language $L$ is $\Sigma^* - L$. In particular, the complement of the empty language is $\Sigma^*$, which is indeed regular.