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Let L be a language and L̄ be its complement. Which one of the following is NOT a viable possibility?

1. Neither L nor L̄ is recursively enumerable (r.e.).
2. One of L and L̄ is r.e. but not recursive; the other is not r.e.
3. Both L and L̄ are r.e. but not recursive.
4. Both L and L̄ are recursive.

Correct answer: Both L and L̄ are r.e. but not recursive.

Solution

Option C is not viable because if both L and its complement L are recursively enumerable but not recursive, it would imply that neither can be decided, leading to a contradiction with the properties of recursive languages.

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