Find the cardinality of $S=\{(x,y,z) \in \Bbb R^3: x^2+y^2=4\}$.
I know that as $S\subseteq \Bbb R^3 \implies |S|\leq \mathfrak{c}$.
My conjecture is that $|S|= \mathfrak c$, I think this is true because the set describes a cylinder in $\Bbb R^3$, and if you (bear with me) unfold this cylinder, you get some kind of rectangle ($\subset$ of some plane) which seems to be equinumerous to a closed rectangle of $\Bbb R^2$.
I'm not sure I made my idea clear. Is there any way to prove this?
The restriction of the identity map in $\Bbb{R}^3$ to $S$ shows gives an injection $S\to\Bbb{R}^3$ so $|S|\le |\Bbb{R}^3|=|\Bbb{R}|$
Then you can create a surjection $S\to \Bbb{R}$ defined by $(x,y,z)\mapsto z$, which shows that $|S|\ge \Bbb{R}$.
Thus $|S|=|\Bbb{R}|$