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Let L1 be a recursive language. Let L2 and L3 be languages that are recursively enumerable but not recursive. Which of the following statements is not necessarily true? (A) L2 – L1 is recursively enumerable. (B) L1 – L3 is recursively enumerable (C) L2 ∩ L1 is recursively enumerable (D) L2 ∪ L1 is recursively enumerable

  • I'm voting to close this question as off-topic because it looks like the author wants to hire someone to prepare their homework. – n. m. could be an AI Mar 27 '16 at 09:34
  • So what have you tried so far to answer this question, and where are you stuck? As written, a plain answer to this plain question is unlikely to help anyone in the future. – Jeroen Mostert Mar 27 '16 at 09:38
  • I think the answer is B. L1 - L3 = L1 intersection ( Complement L3 ) L1 is recursive , L3 is recursively enumerable but not recursive But I'm not sure! –  Mar 27 '16 at 09:57
  • @RiyaK fair enough, you seem to have given the question an honest try. (Do consider accepting the answer if it answered your question. :-) – blazs Mar 27 '16 at 10:17

4 Answers4

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You're right, the answer is (B). You should find a concrete example of languages L1 (a recursive language) and L3 (a RE language) for which L1-L3 is not RE.

Below are proofs that statements (A), (C), and (D) hold. I'm using the fact that every recursive language is recursively enumerable, and the well-known closure properties of RE languages.

(A) L2 - L1 = L2 intersection (complement L1) is recursively enumerable because L1 is recursive, thus (complement L1) is recursively enumerable, and an intersection of RE languages is again a RE language. (In general, the complement of a RE language L is RE if and only if L is recursive.)

(B) L1 - L3 = L1 intersection (complement L3) need not be RE. (Exercise: find a counterexample, i.e., find concrete languages L1 (recursive) and L3 (RE) such that L1-L3 is not RE.)

(C) L2 intersection L1 is RE because both L1 and L2 are RE and we know that intersection of two RE languages is again a RE language.

(D) L2 union L1 is RE because both L1 and L2 are RE and we know that union of two RE languages is again a RE language.

blazs
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L1-L3=L1 intresection need not be RE

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You're right, the answer is (B). You should find a concrete example of languages L1 (a recursive language) and L3 (a RE language) for which L1-L3 is not RE.

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The answer is B.you should find the concrete examples of L1 and L3