Cardinality of Sets

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

and

Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 25 September 2009

Definition

Let $S$ and $T$ be set. $S$ and $T$ have the same cardinality, $Card\left(S\right)=Card\left(T\right)$, if there is a bijective map from $S$ to $T$.

Notation

Let $S$ be a set. Then $$Card\left(S\right)=\left\{\begin{array}{ll}0& \text{if }S=\varnothing \text{,}\\ n& \text{if }Card\left(S\right)=Card\left(\left\{1,2,\dots ,n\right\}\right)\text{,}\\ \mathrm{\infty }& \text{otherwise.}\end{array}\right.$$

Note that even in the case where $Card\left(S\right)=\infty$ and $Card\left(T\right)=\infty$, we may have that $Card\left(S\right)\ne Card\left(T\right)$.

Definitions

A set is $S$ finite if $Card\left(S\right)\ne \infty$.

A set $S$ is infinite if $S$ is not finite.

A set $S$ is countable if either $s$ is finite or $Card\left(S\right)=Card\left({\mathbb{Z}}_{>0}\right)$.

A set $S$ is countably infinite if $S$ is countable and not finite.

A set $S$ is uncountable if $S$ is not countable.

Homework

Show that $Card\left(ℝ\right)=\infty$, that $Card\left(\mathbb{Q}\right)=\infty$ and that $Card\left(ℝ\right)\ne Card\left(\mathbb{Q}\right)$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)