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Cardinal Arithmetic

Definitions

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

$\displaystyle \kappa+\lambda$	$\displaystyle =\vert A\cup B\vert.$
$\displaystyle \kappa\lambda$	$\displaystyle =\vert A\times B\vert.$
$\displaystyle \kappa^\lambda$	$\displaystyle =\vert A^B\vert.$

(Here

denotes the set of all functions from

.) These three operations are well-defined, that is, they do not depend on the choice of

and

. Also note that for multiplication and exponentiation

and

do not actually need to be disjoint.

We also define addition and multiplication for arbitrary numbers of cardinals. Suppose 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 are pairwise disjoint and $\vert A_i\vert=\kappa_i$ for each $i\in I$ . Similarly, $\prod_{i\in I}\kappa_i$ is defined to be the cardinality of the Cartesian product $\prod_{i\in I}B_i$ , where $\vert B_i\vert=\kappa_i$ 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.

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

Some special cases involving 0 and are as follows:

$\displaystyle \kappa+0$	$\displaystyle =\kappa.$
$\displaystyle 0\kappa$	$\displaystyle =0.$
$\displaystyle \kappa^0$	$\displaystyle =1.$
$\displaystyle 0^\kappa$	$\displaystyle =0,$ for $\displaystyle \kappa>0.$
$\displaystyle 1\kappa$	$\displaystyle =\kappa.$
$\displaystyle \kappa^1$	$\displaystyle =\kappa.$
$\displaystyle 1^\kappa$	$\displaystyle =1.$

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

$\displaystyle \kappa+\lambda$	$\displaystyle =\max(\kappa,\lambda).$
$\displaystyle \kappa\lambda$	$\displaystyle =\max(\kappa,\lambda),$ provided $\displaystyle \kappa\ne0\ne\lambda.$

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

$\displaystyle \kappa^\lambda$

$\displaystyle =2^\lambda.$

Inequalities are also important in cardinal arithmetic. The most famous is Cantor's theorem

$\displaystyle \kappa$

$\displaystyle <2^\kappa.$

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

$\displaystyle \mu+\nu$	$\displaystyle \le\kappa+\lambda.$
$\displaystyle \mu\nu$	$\displaystyle \le\kappa\lambda.$
$\displaystyle \mu^\nu$	$\displaystyle \le\kappa^\lambda,$ unless $\displaystyle \mu=\nu=\kappa=0<\lambda.$

Similar inequalities hold for infinite sums and products. Let 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

$\displaystyle \sum_{i\in I}\kappa_i$	$\displaystyle \le\sum_{i\in I}\lambda_i.$
$\displaystyle \prod_{i\in I}\kappa_i$	$\displaystyle \le\prod_{i\in I}\lambda_i.$

If, moreover, $\kappa_i<\lambda_i$ for all $i\in I$ , then we have König's theorem.

$\displaystyle \sum_{i\in I}\kappa_i$

$\displaystyle <\,\prod_{i\in I}\lambda_i.$

If $\kappa_i=\kappa$ for every in the index set , then

$\displaystyle \sum_{i\in I}\kappa_i$	$\displaystyle =\kappa\vert I\vert.$
$\displaystyle \prod_{i\in I}\kappa_i$	$\displaystyle =\kappa^{\vert I\vert}.$

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

1	Real Number
1	Well Defined
1	Disjoint
1	Cardinal Number
1	Union
1	Operation
1	Function
1	Indexing Set
1	Infinite
1	Cardinality
1	Number
2	Generalized Cartesian Product
3	Mutually Disjoint
4	Cantors Theorem
9	Konigs Theorem