Problem: What is the cardinality of all lines $l$ on $\mathbb R^{2}$ which do not contain a point $(x,y)\in l$ where $x, y \in \mathbb Q$ (call it $A$).
My solution: I was thinking of using CB theorem for this problem. It's easy to show that the cardinality of all lines in $\mathbb R^{2}$ is $2^{\aleph_0}$, so it's obvious that $|A|\le 2^{\aleph_0}$, but I'm having trouble of showing that the opposite direction ($|A|\ge 2^{\aleph_0}$). I thought about this injective function ($f:\mathbb R \rightarrow A$)
$\forall r \in \mathbb R$ $f(r)=\left\{\begin{matrix} (r,0), r \in \mathbb R-\mathbb Q & \\ (g(r),0), r \in \mathbb Q& \end{matrix}\right. $
where $g(r) = min{(x\in\mathbb R-\mathbb Q, x \lt r)}$
Is that injective correct? Thanks!