Could someone please provide me with some useful bijections one ought to know for an upcoming examination on cardinality with an emphasis on proofs?
For example, the bijective mapping $f : (-1, 1) \mapsto \mathbb{R}: f(x) = tan(\frac{\pi x}{2})$ can be used to prove that the set of real numbers has the cardinality, $c$ of the continuum etc. And Cantor's diagonalization argument can be used to show that the closed interval $[0, 1]$ is uncountable. Any similar functions that are extremely useful in this regard.
Sometimes maps that make single points "disappear" are useful to have available up your sleeves, e.g. $$\begin{align} f\colon \Bbb R &\to \Bbb R\setminus\{0\}\\x&\mapsto \begin{cases}x+1&x\in\Bbb N_0\\x&\text{otherwise}\end{cases}\end{align}$$ or $$\begin{align} f\colon [0,1] &\to [0,1)\\x&\mapsto \begin{cases}\frac1{\frac 1x+1}=\frac x{x+1}&\frac1x\in\Bbb N\\x&\text{otherwise}\end{cases}\end{align}$$ This can readily be adjusted to make finitely or countably infinitely many points "disappear".