Finding the cardinality of a collection of lines?

Question

I'm trying to find the cardinality of the set of lines such that the lines are not vertical, the slopes are a natural number, and the line has a natural number for a $y$-intercept.

Because there are so many conditions involving natural numbers, I instinctively assumed that the cardinality would be the same as the set of natural numbers? Is this correct at all? How would I mathematically get there?

Answer 1

Each line is given by $y=mx+b$ where $m$ and $b$ are both natural numbers. Moreover, each combination of $m$ and $b$ gives a different line, so there is a natural bijection of this set with $\mathbb N\times\mathbb N$, which is countable.

Answer 2

You can define a map from this set to the set $\Bbb N^2$, namely map each line to the pair made of its slope and its $y$-intercept. This map is injective (in fact, bijective) and of course $\Bbb N^2$ is countable ...