How do I show that the following map establishes a bijection between $\mathbb Q$ and $\mathbb Z \times \mathbb{Z}_{>0}$ Define $$f:\mathbb Z \times \mathbb{Z}_{>0} \to \mathbb{Q}$$ by $$f(p, q) = \frac{p}{q} $$
Edit: Is there an explicit map then that is a bijection?

It doesn't; $f$ is not injective:
$f(2, 4) = \frac{2}{4} = \frac{1}{2} = f(1, 2)$
