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Which of the following statements is/are TRUE?

1. Every subset of a recursively enumerable language is recursive.
2. If a language L and its complement ̅L are both recursively enumerable, then L must be recursive.
3. Complement of a context-free language must be recursive.
4. If L1 and L2 are regular, then L1 ∩ L2 must be deterministic context-free.

Correct answer: If a language L and its complement ̅L are both recursively enumerable, then L must be recursive.

Solution

This statement is true because if both a language and its complement are recursively enumerable, it implies that there exists a Turing machine that can enumerate the strings in both the language and its complement, allowing us to decide membership in the language, thus making it recursive.

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