Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?
L1' --> Complement of L1 L2' --> Complement of L2
(A) L1′ is recursive and L2′ is recursively enumerable
(B) L1′ is recursive and L2′ is not recursively enumerable
(C) L1′ and L2′ are recursively enumerable
(D) L1′ is recursively enumerable and L2′ is recursive
Answer: (B)
Explanation: Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable (Source: http://en.wikipedia.org/wiki/Recursively_enumerable_language )
Recursive Languages are closed under complementation, but recursively enumerable are not closed under complementation. If a languages L is recursively enumerable, then the complement of it is recursively enumerable if and only if L is also recursive. Since L2 is recursively enumerable, but not recursive, L2′ is not recursively enumerable.
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