Calculate the cardinal numbers of the following subsets of $\mathbb R\times\mathbb R$ :
a.$X=\left\{ (a,b)\in\mathbb{R}\times\mathbb{R}\mid a+b\in\mathbb{Q}\right\} $
b.$Y=\left\{ (a,b)\in\mathbb{R}\times\mathbb{R}\mid a\in\mathbb{Q}\wedge b^2\in\mathbb Q\right\} $
About X: I think $|X|=\aleph_0$ but the only function I can think of is $f:\mathbb {R\times R}\to\mathbb Q$ defined by $(a,b)\to a+b$, which is surjective but not injective. Maybe the cardinal is aleph? I so,what's the general intuition for it?
about Y:I think $|Y|=\aleph_0$ but again, I have troubles define appropriate function.
How can I define the 'right' functions?
HINT: For $X$ note that $x+(-x)\in\Bbb Q$ for all $x\in\Bbb R$. To prove that $|Y|=\aleph_0$, it suffices to prove that $\{b\in\Bbb R:b^2\in\Bbb Q\}$ is countable.