 Solution

The integers are countable under the enumerationPart (i) then shows that $\mathbb{Z}\times \mathbb{N}$ is countable. The mapis a surjection from $\mathbb{Z}\times \mathbb{N}$ onto the rational numbers. Selecting, for example, the representation in lowest terms with positive denominator gives an injection in the other direction. Thus $\mathbb{Q}$ is countable.