A is the family of all circles in the plane $\Bbb R^2$. They all have rational centers and rational radius. Show that A is countable.
Here is my reasoning:
Given the details, the set $ A = \{(x-h)^2+(y-k)^2=r^2:x,y,h,k,r \in\Bbb Q \}$
Since, we know, $\Bbb Q $ is countable and $A \subset \Bbb Q $ it implies that $A$ is countable
I think this is too simple and it probably has to do with showing $\Bbb Q \times \Bbb Q \times \Bbb Q \times\Bbb Q \times\Bbb Q $ is countable
I'm a bit stuck. Any help or references to material I can read to understand this would be great!
A set $S$ is countable iff there is some injection $i_S : S \to \mathbb{N}$.
So if $A \subseteq S$ and $S$ is countable, the injection $i_S : S \to \mathbb{N}$ restricts to an injection $i_S|_A : A \to \mathbb{N}$, and hence $A$ is countable.