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

Question

I have this problem:

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

I know that $\mathbb Q$ is countably infinite.

But then how can I prove that $\mathbb{|Q\times Q|}$ is countably infinite?

Thanks you!

Answer 1

Whatever proof you have that $\mathbb Q$ is countably infinite probably relies on a mapping between elements of $\mathbb Q$ and elements of $\mathbb {Z \times Z}$. But to say that $\mathbb Q$ is countably infinite is to put it in correspondence with $\mathbb Z$. Use this fact, then repeat the original proof.

Answer 2

Use $\mathbb{|N\times N|=|N|}$ and $\mathbb{|N|=|Q|}$.