I am given two sets:
$$A= \{ (x,y)\in \mathbb{Q}^2 \mid x-y \in \mathbb{Z} \}$$
$$B= \{ (x,y)\in \mathbb{R}^2 \mid x-y \in \mathbb{Q} \}$$
I am told that $A$ is countable and $B$ is not. For a set to be countable I need to show that the cardinality of that set is less than or the same as the one for the natural numbers. If this is not the case it is uncountable.
If I could find a injection from A into $\mathbb{N}$ I am done. If I can show that no injection exists from $B$ into $\mathbb{N}$ I am done. However, this method does not seem to be that obvious here.
I know that $\mathbb{R}^2$ is uncountable and $\mathbb{Q}^2$ is countable as well as $\mathbb{Z}$ but do not know how to proceed.
How do I formally show that $A$ is countable but $B$ is not (I am really interested in the method/approach)?
$\mathbb{Q}^2$ itself is countable. Since $A\subset\mathbb{Q}^2$, It should also be countable.
We have the subset $H=\{(x,x)\mid x\in\mathbb{R}\}\subset B$. $H$ is clearly uncountable since you have trivial bijection from $H$ to $\mathbb{R}$. This proves that $B$ is uncountable as well.