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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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).