Cardinal

Intuitive definition

The cardinal of a finite set $E$ is the number of elements in $E$. It is often denoted as $\vert E\vert$ or $\#E$ or also $\mathrm{Card}(E)$.

Formal definition

More formally, the cardinal of a set $E$ is the smallest integer $n$ such that there exists an injection from $E$ to $[\![1,n]\!]$. If it does not exists, we set $\vert E\vert=+\infty$.

Examples

$\bullet$ The set $\{1,8,36\}$ has $3$ elements, hence $\vert \{1,8,36\}\vert=3$.
$\bullet$ The set $\{(1,2), \text{Anna}, \pi^2, \{1\}\}$ has $4$ elements, hence its cardinal is $4$.
$\bullet$ There is an infinite number of integers, so $\vert \Z\vert=+\infty$.