GATE | GATE-CS-2016 (Set 1) | Question 54

Let X be a recursive language and Y be a recursively enumerable but not recursive language. Let W and Z be two languages such that Y’ reduces to W, and Z reduces to X’ (reduction means the standard many-one reduction). Which one of the following statements is TRUE
(A) W can be recursively enumerable and Z is recursive.
(B) W an be recursive and Z is recursively enumerable.
(C) W is not recursively enumerable and Z is recursive.
(D) W is not recursively enumerable and Z is not recursive

Answer: (C)

Explanation: Since X is recursive and recursive language is closed under complement. So X’ is also recursive.
Since  Z ≤ X’ is recursive. (Rule : if Z is reducible to X’ , and X’ is recursive, then Z is recursive.)
Option (B) and (D) is eliminated.
And Y is recursive enumerable but not recursive, so Y’ cannot be recursively enumerable.
Since Y’ reduces to W.
And we know complement of recursive enumerable is not recursive enumerable and therefore, W is not recursively enumerable. So Correct option is (C).

Here Y’ is complement of Y
and X’ is complement of X.


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