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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
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let X be a recursive language and Y be a recursively enumerable but no...
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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Most Upvoted Answer
Let X be a recursive language and Y be a recursively enumerable but no...
Explanation:

Definitions:
- Recursive language: A language for which there exists a Turing machine that halts on all inputs and accepts a string if it belongs to the language.
- Recursively enumerable language: A language for which there exists a Turing machine that halts on all inputs and accepts a string if it belongs to the language, but may loop indefinitely on inputs not in the language.
- Many-one reduction: A mapping from instances of one decision problem to instances of another such that a solution to the second problem can be used to solve the first.

Analysis:
- Y reduces to W: This means that there is a computable function that transforms instances of Y into instances of W, allowing solutions to W to solve Y.
- Z reduces to X: This means that there is a computable function that transforms instances of Z into instances of X, allowing solutions to X to solve Z.

Explanation of True Statement:
- W is not recursively enumerable: Since Y reduces to W, and Y is recursively enumerable but not recursive, W cannot be recursively enumerable. If W were recursively enumerable, then Y would also be recursive, leading to a contradiction.
- Z is recursive: Since Z reduces to X, and X is recursive, Z must also be recursive. If Z were not recursive, then X would not be recursive, contradicting the given information.
Therefore, the true statement is:

W is not recursively enumerable and Z is recursive (option c).
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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 TRUEa)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 recursiveCorrect answer is option 'C'. Can you explain this answer?
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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 TRUEa)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 recursiveCorrect answer is option 'C'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about 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 TRUEa)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 recursiveCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 TRUEa)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 recursiveCorrect answer is option 'C'. Can you explain this answer?.
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