Cardinality of a finite set

Definition

Let $A$ be a set and let $n$ be a natural number. The set $A$ has $n$ elements if there is a bijection from $\{1, 2, \dots, n\}$ to $A$. For $n = 0$, the set $\{1, \dots, n\}$ is the empty set, so only the empty set has $0$ elements. A set is finite if it has $n$ elements for some natural number $n$.

For a finite set $A$, this number $n$ is unique. It is the cardinality of $A$, with the notation $|A|$ or $\# A$.

Remarks

Two finite sets have the same cardinality if and only if there is a bijection between them. The bijection principle states this. For example, $|\{a, b, c\}| = 3$ and $|\varnothing| = 0$. The uniqueness of $n$ follows from the fact that there is no injection from $\{1, \dots, m\}$ to $\{1, \dots, n\}$ when $m > n$.

Explanations

L1 human

The cardinality of a finite set is the number of its elements. To count a set, you label its elements $1, 2, 3, \dots$ without gaps and without repeats. The last label is the cardinality.