# cardinal arithmetic

Cardinal arithmetic is distinct from ordinal arithmetic in the case of transfinite numbers. Let $$\alpha$$ and $$\beta$$ denote cardinals and let $$A$$ and $$B$$ denote disjoint sets such that $$\alpha = \left| A \right|$$ and $$\beta = \left| B \right|$$. Denote by $$A^B$$ the set of all functions from $$B$$ to $$A$$, and by $$A\times B$$ the Cartesian product of $$A$$ and $$B$$. The operations of cardinal arithmetic are then defined by
• $$\alpha + \beta = \left| A \cup B \right|$$,
• $$\alpha \beta = \left|A \times B\right|$$, and
• $$\alpha^{\beta} = \left|A^B\right|$$.

Note for instance that if $$A$$ or $$B$$ is infinite then $$\alpha + \beta$$ is just the larger of $$\alpha$$ or $$\beta$$.