07 - Countability of Positive Rationals

Countability of Positive Rationals

$${\mathbb{Q}}^{>0}\text{ is countable}$$

Proof

Define $f:{\mathbb{Q}}^{>0}\to \mathbb{N}$ by

$$f\left(\frac{p}{q}\right)=2^p{3}^q$$

where $p,q\in {\mathbb{Z}}^{>0}$ and $\mathrm{hcf}(p,q)=1$
And this is also an injection