Cardinality

Definition · applies to sets · also called $\abs{A}$, $\#A$

Elementary

The cardinality of a set is how many elements it has.

Caveat. For infinite sets, “how many” needs a new idea: two sets have the same cardinality if their elements can be paired up one-to-one.

High school

The cardinality of a finite set is its number of elements.

Graduate (set theory)

Sets $A$ and $B$ have the same cardinality if there is a bijection between $A$ and $B$. This extends counting to infinite sets.

Examples

The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$ : $\mathfrak{c}$ High school
The cardinality of the closed unit disk is $\mathfrak{c}$, the same as that of $\reals$.
$[0, 1]$ : $\mathfrak{c}$ High school
The cardinality of $[0, 1]$ is $\mathfrak{c}$, the same as that of $\reals$, because Cantor’s diagonal argument shows that $[0, 1]$ is uncountable, and $[0, 1]$ is in bijection with $\reals$.
$\mathbb{Q} \cap [0, 1]$ : $\aleph_0$ High school
The cardinality of $\mathbb{Q} \cap [0, 1]$ is $\aleph_0$, because its elements can be listed in order of their denominators.
The set of real numbers, $\reals$ : $\mathfrak{c}$ High school
The cardinality of $\reals$ is $\mathfrak{c}$, the continuum, which is uncountable.
The set $\setdef{0, 1}$, as a subset of $\reals$ : $2$ High school
The cardinality of $\setdef{0, 1}$ is $2$, because it has $2$ elements.
$\setdef{1, 2, 3}$ : $\abs{\setdef{1, 2, 3}} = 3$ High school
The Cantor set : $\mathfrak{c}$ Undergraduate
The cardinality of the Cantor set is $\mathfrak{c}$, because its points correspond to the infinite sequences of $0$s and $2$s (their base-$3$ digits).