---
tag: 0JTGRV4WT4
kind: definition
name: Cardinality
msc: ["03E10"]
defines: 0G0ASKNA6Z
uses: [08EJHHMB5R]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinality
    locator: lead section
---

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The cardinality of $A$, written $|A|$, is an object assigned to $A$ such that, for all sets $A$ and $B$,
$$|A| = |B| \iff A \text{ and } B \text{ are } \text{[[equinumerous|08EJHHMB5R]]}.$$
For a finite set $A$, $|A|$ is the number $n$ of its elements, that is, the unique $n \in \mathbb{N}$ with $A \approx \{1, \dots, n\}$.
For an infinite set, $|A|$ is a [[cardinal number|0QKPNZKNEA]].

## Remarks

Other notations are $\#A$, $\operatorname{card}(A)$ and $\overline{\overline{A}}$.
With the [[axiom of choice|0HTGB13FE2]], one can take $|A|$ to be the least [[ordinal|0RDJK7EQYY]] equinumerous with $A$.
$|A| \le |B|$ means that there is an [[injection|0PVG1CRAJN]] from $A$ to $B$. See [[order of cardinals|01GTP8M5RG]].

## Explanations

### L1 human

The cardinality of $\{a, b, c\}$ is $3$.
For infinite sets, cardinality is a new kind of number that records which sets can be put in bijection.
For example, $|\mathbb{N}| = |\mathbb{Z}|$, but $|\mathbb{N}| < |\mathbb{R}|$.