Countable set of number rational, prove with $\mathbb{Z}$.

Question

Good morning, I need to prove $ \mathbb{Q} $ is a countable set, but I prove $ \mathbb{Z} $ is a countable set, now, can I use this for proving $ \mathbb{Q} $ is countable set? I was thinking about a bijective function with $ \mathbb{Z} $ and $ \mathbb{Q} $ for say $ \mathbb{Q} $ is countable set.

Answer 1

Here is the classic

Put the natural numbers on the horizontal and vertical axes of a grid.

Filling the grid, take integer on the horizontal axis, and divide it by the integer on the vertical axis.

The grid is filled with the positive rational numbers. Yes many numbers are duplicated, but every rational is represented.

Now draw a serpentine line that runs diagonally across the grid, and ever time it gets to the edge wind back along the next diagonal. i.e. Start at $(1,1)$ proceeding to $(1,2), (2,1), (3,1) \to (1,3), (1,4) \to (4,1)$, etc.

every rational number has now been mapped to a position on this line and a 1-1 correspondence can be established with $\mathbb N$

The positive rational numbers are countable. It is a quick jump to all rational numbers are countable.

Answer 2

$ f(\pm\frac{a}{b}) = \pm2^a 3^b $ is an injection $ \mathbb{Q} \to \mathbb{Z} $.