GATE CS 2014 Set 1 — Question 45
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Theory of Computation → Turing Machines & Computability → Recursive & RE Languages
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Question
Let be a language and be its complement. Which one of the following is NOT a viable possibility?
Correct answer
(C) Both L and L are r.e. but not recursive.
Solution
A standard theorem in theory of computation states that if a language and its complement are both recursively enumerable (r.e.), then is recursive (decidable). Therefore, it is impossible for both to be r.e. but NOT recursive. Option (C) describes this impossible scenario.
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