Prove that $\mathbb{|Q|} = \mathbb{|Q \times Q|} $

Question

This question exists, but both cases have a specific answer for the OP's situation. I do not know how to prove that $\mathbb{Q}$ is countable.

Questions I am referring to:
Prove that $\mathbb{|Q| = |Q\times Q|}$ and Prove that $\mathbb{Q} \times \mathbb{Q}$ is countable.

Answer 1

Hint: You can check that

$$\varphi(\frac{p}{q},\frac{r}{s})=\frac{2^p3^q}{5^r7^s}$$

where $p$ and $q$ are integers with no common factor and $q$ is positive, and similarly for $r$ and $s$, is an injection of ${\mathbb Q}\times{\mathbb Q}$ into ${\mathbb Q}$.

On a side note, if you study about cardinals of sets, it is a good idea to prove that ${\mathbb Q}$ is countable, sooner or later.