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# 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 $$\beta$$.

Citation Info

• [MLA] “cardinal arithmetic.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 20 Mar 2013. Web. 20 Mar 2013. <http://platonicrealms.com/>
• [APA] cardinal arithmetic (20 Mar 2013). Retrieved 20 Mar 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/cardinal-arithmetic/