Prove that the set of all straight lines, each of which passes through at least two point such that its two coordinates are integers, is Countable set.
If $A$ is a set of straight lines referred to in the task, then I think $$A=\left\{ y=ax+b, \ a,b\in \mathbb C \right\} \cup \left\{ x=c, \ c\in\mathbb C \right\} $$ and exist at least two $a,b \in \mathbb Z$ or $c \in \mathbb Z$.
However I don't knew how I can show that $A$ is countable.
Can anyone help me?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The set$$S=\left\{\bigl((a,b),(c,d)\bigr)\in\mathbb{Z}^2\times\mathbb{Z}^2\,\middle|\,(a,b)\neq(c,d)\right\}$$is countable. For each element of $\bigl((a,b),(c,d)\bigr)\in S$, there is one and only one line passing through $(a,b)$ and $(c,d)$. This defines a surjective map from $S$ onto your set. So, since $S$ is countable, your set is either finite or countable. But it is clearly infinite. So…
answered Jan 8, 2019 at 18:04