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A theorem says if $f$ is a computable function and we can prove $x \in A \Leftrightarrow f(x) \in B$, then we can use reduction so $A \leq_m B$.

But i'm confused if should I prove if :

  1. $(x \in A \Rightarrow f(x) \in B )\land (f(x) \in B \Rightarrow x \in A)$

Or

  1. $(x \in A \Rightarrow f(x) \in B ) \land(x \notin A \Rightarrow f(x) \notin B)$

in which circunstances because I've seen both in demonstration. Or maybe are they equivalent ?

For example, here is a solution to the empty string problem. But with $f$ such as, $f: \text{M accepts w} \to \text{M accepts } \epsilon \\ f(\langle M, \epsilon \rangle) = \langle M \rangle$, we can proove 1. Is it valid ?

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It is called contraposition :

$f(x) \in B \Rightarrow x \in A$

is equivalent to

$x \notin A \Rightarrow f(x) \notin B$.

So, the two statements that you have seen are equivalent as well.

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