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 \(\beta\).