Definition 1. The sum of two cardinals $\kappa$ and $\lambda$ is
\[\kappa + \lambda = \lvert (\kappa \times \{0\}) \cup (\lambda \times \{1\}) \rvert.\]Proposition 1. For any two disjoint sets $X$ and $Y$,
\[\lvert X \rvert + \lvert Y \rvert = \lvert X \sqcup Y \rvert.\]Definition 2. The product of two cardinals $\kappa$ and $\lambda$ is
\[\kappa \cdot \lambda = \lvert \kappa \times \lambda \rvert.\]Proposition 2. For any two sets $X$ and $Y$,
\[\lvert X \rvert \cdot \lvert Y \rvert = \lvert X \times Y \rvert.\]Definition 3. The exponentiation of a cardinal $\kappa$ to the power of another cardinal $\lambda$ is
\[\kappa^\lambda = \lvert \fun(\lambda, \kappa) \rvert.\]Proposition 3. For any two sets $X$ and $Y$,
\[\lvert X \rvert^{\lvert Y \rvert} = \lvert \fun(Y, X) \rvert.\]