Show that \(\mathbb{Q}\) is countably infinite.
Solution
The positive rationals are countable. Indeed, the map \[ q:\mathbb Z_+\times\mathbb Z_+\longrightarrow\mathbb Q_+, \qquad q(m,n)=\frac mn, \] is surjective, and \(\mathbb Z_+\times\mathbb Z_+\) is countable. Hence \(\mathbb Q_+\) is countable.
Now \[ \mathbb Q=\mathbb Q_+\cup\{0\}\cup(-\mathbb Q_+), \] a finite union of countable sets, so \(\mathbb Q\) is countable. It is infinite because it contains \(\mathbb Z_+\). Therefore \[ \boxed{\mathbb Q\text{ is countably infinite}.} \]