# Condition to prove $f$ is a reduction

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 ?

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