# cardinal arithmetic

## Definitions

Let $\kappa $ and $\lambda $ be cardinal numbers^{},
and let $A$ and $B$ be disjoint sets such that $|A|=\kappa $ and $|B|=\lambda $.
(Here $|X|$ denotes the cardinality of a set $X$,
that is, the unique cardinal number equinumerous with $X$.)
Then we define cardinal addition^{}, cardinal multiplication
and cardinal exponentiation as follows.

$\kappa +\lambda $ | $=|A\cup B|.$ | ||

$\kappa \lambda $ | $=|A\times B|.$ | ||

${\kappa}^{\lambda}$ | $=|{A}^{B}|.$ |

(Here ${A}^{B}$ denotes the set of all functions from $B$ to $A$.)
These three operations^{} are well-defined, that is,
they do not depend on the choice of $A$ and $B$.
Also note that for multiplication and exponentiation $A$
and $B$ do not actually need to be disjoint.

We also define addition and multiplication for arbitrary numbers of cardinals.
Suppose $I$ is an index set^{} and ${\kappa}_{i}$ is a cardinal for every $i\in I$.
Then ${\sum}_{i\in I}{\kappa}_{i}$ is defined to be
the cardinality of the union ${\bigcup}_{i\in I}{A}_{i}$,
where the ${A}_{i}$ are pairwise disjoint and $|{A}_{i}|={\kappa}_{i}$ for each $i\in I$.
Similarly, ${\prod}_{i\in I}{\kappa}_{i}$ is defined to be the cardinality of the
Cartesian product (http://planetmath.org/GeneralizedCartesianProduct)
${\prod}_{i\in I}{B}_{i}$, where $|{B}_{i}|={\kappa}_{i}$ for each $i\in I$.

## Properties

In the following, $\kappa $, $\lambda $, $\mu $ and $\nu $ are arbitrary cardinals, unless otherwise specified.

Cardinal arithmetic obeys many of the same algebraic laws as real arithmetic^{}.
In particular, the following properties hold.

$\kappa +\lambda $ | $=\lambda +\kappa .$ | ||

$(\kappa +\lambda )+\mu $ | $=\kappa +(\lambda +\mu ).$ | ||

$\kappa \lambda $ | $=\lambda \kappa .$ | ||

$(\kappa \lambda )\mu $ | $=\kappa (\lambda \mu ).$ | ||

$\kappa (\lambda +\mu )$ | $=\kappa \lambda +\kappa \mu .$ | ||

${\kappa}^{\lambda}{\kappa}^{\mu}$ | $={\kappa}^{\lambda +\mu}.$ | ||

${({\kappa}^{\lambda})}^{\mu}$ | $={\kappa}^{\lambda \mu}.$ | ||

${\kappa}^{\mu}{\lambda}^{\mu}$ | $={(\kappa \lambda )}^{\mu}.$ |

Some special cases involving $0$ and $1$ are as follows:

$\kappa +0$ | $=\kappa .$ | ||

$0\kappa $ | $=0.$ | ||

${\kappa}^{0}$ | $=1.$ | ||

${0}^{\kappa}$ | $=0,\text{for}\kappa 0.$ | ||

$1\kappa $ | $=\kappa .$ | ||

${\kappa}^{1}$ | $=\kappa .$ | ||

${1}^{\kappa}$ | $=1.$ |

If at least one of $\kappa $ and $\lambda $ is infinite^{}, then the following hold.

$\kappa +\lambda $ | $=\mathrm{max}(\kappa ,\lambda ).$ | ||

$\kappa \lambda $ | $=\mathrm{max}(\kappa ,\lambda ),\text{provided}\kappa \ne 0\ne \lambda .$ |

Also notable is that if $\kappa $ and $\lambda $ are cardinals with $\lambda $ infinite and $2\le \kappa \le {2}^{\lambda}$, then

${\kappa}^{\lambda}$ | $={2}^{\lambda}.$ |

Inequalities are also important in cardinal arithmetic.
The most famous is Cantor’s theorem^{}

$\kappa $ | $$ |

If $\mu \le \kappa $ and $\nu \le \lambda $, then

$\mu +\nu $ | $\le \kappa +\lambda .$ | ||

$\mu \nu $ | $\le \kappa \lambda .$ | ||

${\mu}^{\nu}$ | $$ |

Similar inequalities hold for infinite sums and products^{}.
Let $I$ be an index set,
and suppose that ${\kappa}_{i}$ and ${\lambda}_{i}$ are cardinals for every $i\in I$.
If ${\kappa}_{i}\le {\lambda}_{i}$ for every $i\in I$, then

$\sum _{i\in I}}{\kappa}_{i$ | $\le {\displaystyle \sum _{i\in I}}{\lambda}_{i}.$ | ||

$\prod _{i\in I}}{\kappa}_{i$ | $\le {\displaystyle \prod _{i\in I}}{\lambda}_{i}.$ |

If, moreover, $$ for all $i\in I$, then we have König’s theorem.

$\sum _{i\in I}}{\kappa}_{i$ | $$ |

If ${\kappa}_{i}=\kappa $ for every $i$ in the index set $I$, then

$\sum _{i\in I}}{\kappa}_{i$ | $=\kappa |I|.$ | ||

$\prod _{i\in I}}{\kappa}_{i$ | $={\kappa}^{|I|}.$ |

Thus it is possible to define exponentiation in terms of multiplication, and multiplication in terms of addition.

Title | cardinal arithmetic |

Canonical name | CardinalArithmetic |

Date of creation | 2013-03-22 14:15:13 |

Last modified on | 2013-03-22 14:15:13 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 38 |

Author | yark (2760) |

Entry type | Topic |

Classification | msc 03E10 |

Related topic | OrdinalArithmetic |

Related topic | CardinalNumber |

Related topic | CardinalExponentiationUnderGCH |

Related topic | CardinalityOfTheContinuum |

Defines | cardinal addition |

Defines | cardinal multiplication |

Defines | cardinal exponentiation |

Defines | sum of cardinals |

Defines | product of cardinals |

Defines | addition |

Defines | multiplication |

Defines | exponentiation |

Defines | sum |

Defines | product |