Proving that a set is countable (or not) Prove that the set $$\left \{ (u,v)\in\mathbb{Q}^2:u-v\in \mathbb{Z}  \right \}$$ is indeed countable and that
$$\left \{ (u,v)\in\mathbb{R}^2:u-v\in \mathbb{Q}  \right \}$$
is not

By instinct I would say the first set is countable as $u-v\in \mathbb{Z} $ and that the rationals are countable (and thus the cart. prod.) but how does one rigorously show that a set is either countable or uncountable?
I know that a set $S$ is countable if its cardinality is less than or equal to $\aleph_0$.
The second one instinctly is also not countable to me as the elements are from $\mathbb{R}^2$.
 A: For the first one remember that both $\mathbb Q$ and $\mathbb Q^2$ are countable, and with $q\to(q,q)$ there is an injection from $\mathbb Q$ into the set. Thus the set is countable as well.
For the second one: We again have that $x\to(x,x)$ injects $\mathbb R$ into your set. So the set cannot be countable (as it contains an injective image of $\mathbb R$.
A: First, there is an ambiguity about countable, which some use to mean countably infinite and others to include finite sets as well. I generally include finite sets because it allows me to say that a subset of a countable set is countable. And that is useful to prove a set countable - because we might be able to see that it is a subset of a known countable set.
For proving a set uncountable, the same idea allows us to show that some subset of the original is uncountable.
These observations, like the fact that the Cartesian product of countable sets is countable, become part of a toolkit which enables us to prove that sets are countable or uncountable without going back to absolute basics every time, and by reference to a known collection of countable and uncountable sets.
To use this principle in your toolkit, you first have to prove it as rigorously as required in your context.
