Cardinality¶

Definition¶

Two sets $A$ and $B$ have the same cardinality, written

\operatorname{card}(A)=\operatorname{card}(B),

if there exists a bijection $f\colon A\to B\text{.}$

The empty set has cardinality $0\text{.}$ A set $A$ has finite cardinality $n$ if it has the same cardinality as $\{1,2,\ldots,n\}\text{.}$

Interpretation¶

Cardinality measures the size of a set through one-to-one correspondence. For finite sets this reproduces ordinary counting; for infinite sets it provides a precise way to compare sizes.

Two sets A and B with three elements each, joined in pairs by lines that form a bijection, showing that they have the same cardinality.

Each element of $A$ is joined to exactly one element of $B$ , and every element of $B$ is reached once, so the lines form a bijection and $\operatorname{card}(A)=\operatorname{card}(B)=3$ $3$ . No counting is needed to see that the sets are equally large: the pairing itself shows it, just as a room has as many chairs as people when everyone sits on a chair and no chair is empty.

Because the definition compares sets instead of counting them, it also works for infinite sets, with some surprising answers. $\mathbb{N}$ and the even natural numbers have the same cardinality, although one is a proper subset of the other, because $n\mapsto2n$ pairs them. To show that two sets do not have the same cardinality, one failed attempt is not enough: one must show that no bijection exists at all, as Cantor's diagonal argument does for $\mathbb{N}$ and $\mathbb{R}$ .

The infinite sizes have their own names: aleph null, $\aleph_0$ , for the countably infinite sets such as $\mathbb{N}$ , $\mathbb{Z}$ and $\mathbb{Q}$ , and the cardinality of the continuum, $\mathfrak{c}$ , for $\mathbb{R}$ .

Key Results¶

Having equal cardinality is an equivalence relation on sets.
A finite set with $n$ distinct elements has cardinality $n\text{.}$
Infinite sets can have the same cardinality even when one is a proper subset of the other.
Cantor's theorem shows that every set has strictly smaller cardinality than its power set; here $\operatorname{card}A<\operatorname{card}B$ means that there is an injection $A\to B$ but no bijection.

Examples¶

$\{1,2,3\}$ and $\{a,b,c\}$ have the same cardinality.
$\mathbb{N}$ and the even natural numbers have the same cardinality because $n\mapsto 2n$ is a bijection.
$\mathbb{N}$ , $\mathbb{Z}$ , and $\mathbb{Q}$ are countably infinite, whereas $\mathbb{R}$ is uncountable.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: equinumerous, equipotent, cardinal number Symbols: integers, natural numbers, rational numbers, real numbers

Last updated: 20 October 2026 03:02