Hint: Fix $a_n$ as a sequence of irrational numbers, and write $\mathbb Z=\{z_n\mid n\in\mathbb N\}$. Define a function which sends $a_n$ to $a_{2n}$; $z_n$ to $a_{2n+1}$; and $x$ to itself otherwise.
One domain is $\{\rho_1(x,y)<1\}$, the other is $\{\rho_2(x,y) <1\}$, where $\rho_1(x,y) =\sqrt{x^2+y^2}$, and $\rho_2(x,y) = \max(|x|,|y|)$. Find a map $F\colon (x,y) \mapsto (x',y')$ so that $\rho_1(x,y) = \rho_2(x',y')$. It should be linear on each line through the origin. One can take:
$$F(x,y) = \frac{\rho_1(x,y)}{\rho_2(x,y)} \cdot (x,y)$$
that is
\begin{eqnarray}
F(x,y) = \frac{\sqrt{x^2+y^2}}{\max(|x|,|y|)} \cdot (x,y)
\end{eqnarray}
with inverse
\begin{eqnarray}
F^{-1}(x,y) = \frac{\max(|x|,|y|)}{\sqrt{x^2+y^2}} \cdot (x,y)
\end{eqnarray}
$F$ provides a homeomorphism from the disk to the square.
Best Answer
Hint 1: There are obvious inclusions $$\mathbb{N} \hookrightarrow \mathbb{Q}_{\ge 0} \hookrightarrow \mathbb{Q}$$ so if you can find an injection $\mathbb{Q} \to \mathbb{N}$ then you'll have proved that all three sets have the cardinality.
Hint 2: (hover mouse over to see)
Hint 3: (hover mouse over to see)