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0. Let A and B be infinite alphabets and let # be a symbol outside both A and B. Let f be a total functional from A* to B*. We say f is computable if there exists a Turning machine M which given an input x in A*, always halts with f(x) on its tape. Let Lf denote the language {x ≠ f(x) | x ∈ A*}. Which of the following statements is true:
f if computable if and only if Lf is recursive.
f is computable if and only Lf recursively enumerable.
If f is computable then Lf is recursive, but not conversely.
If f is computable then Lf is recursively enumerable, but not conversely.
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