$A =\{x\in R: x>0 \land x^2 \in Q \}$
How do I prove this set to be countably infinite? I cannot come up with a function which maps rationals to the numbers of this set. Where do I start from?
I think I should start from $\sqrt{2}$ and move on either side of it. But that doesn't sound like a rigorous argument.
Do I need to think in terms of Dedekind cuts?
Thank you! Any help would be greatly appreciated.
You have that $\mathbb{Q}\subset A$.
And that $f:A\to \mathbb{Q}$ given by $f(x)=x^2$ is injective. In fact, if $x,y\in A$ and $x^2=y^2$. Then, one of them is larger. Assume it is $x\geq y$. Then $(x-y)(x+y)=x^2-y^2=0$. Therefore, $x-y=0$. Note that we cannot have $x+y=0$, since $x,y>0$.
Then $x=y$.
So, you have an injection from $\mathbb{Q}$ to $A$ and an injection from $A$ to $\mathbb{Q}$. Schroeder-Bernstein implies that there is a bijection between $A$ and $\mathbb{Q}$.