Prove that the sets $S$ and $D$ have the same cardinality

Question

Prove that the sets $S$ and $D$ have the same cardinality, where $S = \{(x,y)\mid-1\leq x \leq 1\text{ and }-1\leq y\leq 1\}$ and $D = \{(x,y)\mid x^2 + y^2 \leq 1\}$.

Answer 1

HINT: Both these sets are subsets of $\Bbb R^2$, whose cardinality is the same as that of $\Bbb R$. It suffices to find subsets of both these sets (or even a subset of $D\cap S$) whose cardinality is also the same as $\Bbb R$.

Answer 2

Hint: Define the map $f:[-1,1]^2\to D$ such that $$f(x,y) = \frac{\max\{|x|,|y|\}}{\sqrt{x^2+y^2}}(x,y), $$ then prove this map is one-to-one and onto!