Determining the cardinality of these sets.

Question

I am having trouble with determining the cardinality(finite, denumerable, uncountable) of these two sets:

1. The set of all circles in $\mathbb{R}^2$ in form $(x-a)^2+(y-b)^2=R^2$ with $a,b,R\in\mathbb{Q}$.

2. $\{(x,y)\in\mathbb{R}^2:x+y\in\mathbb{Q}\}$

Work:I think that both sets are uncountable since they are both infinite subsets of $\mathbb{R}^2$ which is an uncountable set.

Answer 1

Hint: If you don't already have it as a theorem, prove that ${\mathbb N} \times {\mathbb N}$ is countable, and then you will get that the first set is countable. For the second set, consider all pairs of the form $(x,1-x)$ where $x$ is irrational, and show this set is uncountable.

Answer 2

HINT: Note that circle is uniquely determined by $a,b,R$. So there is a natural bijection between the first set and a subset of $\Bbb Q^3$. For the second set, note that for every $x\in\Bbb R$ there is some $y$ such that $x+y=0$.