Suppose there are four languages A, B, C, and D. Each of the languages may or may not be recursively enumerable. However, we know the following about them:

• There is a reduction from A to B.
• There is a reduction from B to C.
• There is a reduction from D to C.

Which of the following option can never be true?
(A) A is recursively enumerable but not recursive, and C is recursive.
(B) A is not recursive, and D is not recursively enumerable.
(C) If C is recursive, then the complement of D is recursive.
(D) If C is recursively enumerable, then B ∩ D is recursively enumerable.


Answer: (A)

Explanation: (A) NEVER TRUE. reductions are transitive, and since A reduces to B, and B reduces to C, we conclude that A reduces to C. Therefore, if A ∉ R, it can’t be that C ∈ R.

(B) MAYBE TRUE.

(C) CERTAIN to be TRUE. If C is in R, and since D reduces to C, then by the reduction theorem, it follows that D is in R. Since R is closed under complement, then the complement of D is also in R.

(D) CERTAIN to be TRUE. C is in RE implies that both B and D are in RE (by the reduction theorem), and so is their intersection (RE langs are closed under intersection).


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