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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?

1. W can be recursively enumerable and Z is recursive.
2. W can be recursive and Z is recursively enumerable.
3. W is not recursively enumerable and Z is recursive.
4. W is not recursively enumerable and Z is not recursive.

Correct answer: W is not recursively enumerable and Z is recursive.

Solution

Since Y is r.e. but not recursive, its complement bar-Y is not r.e.; as bar-Y many-one reduces to W, W cannot be r.e. either. X is recursive so bar-X is recursive, and Z reducing to bar-X makes Z recursive. Thus W is not r.e. and Z is recursive.

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