For any cardinals $\kappa$ and $\lambda$,
\[\kappa+\lambda = \lvert(\kappa\times\{0\})\cup(\lambda\times\{1\})\rvert.\]
If $X$ and $Y$ are disjoint,
\[\lvert X\rvert+\lvert Y\rvert = \lvert X\cup Y\rvert.\]
For any cardinals $\kappa$ and $\lambda$,
\[\kappa\cdot\lambda = \lvert\kappa\times\lambda\rvert.\]
For any $X$ and $Y$,
\[\lvert X\rvert\cdot\lvert Y\rvert = \lvert X\times Y\rvert.\]
For any cardinals $\kappa$ and $\lambda$,
\[\kappa^\lambda = \lvert\fun(\lambda,\kappa)\rvert.\]
For any $X$ and $Y$,
\[\lvert X\rvert^{\lvert Y\rvert} = \lvert\fun(Y,X)\rvert.\]