= Cardinality
{title2=$|X|$}
{wiki}
The cardinality $|X|$ of a <set> $X$ expresses its size up to <bijection>. Two <sets> have the same cardinality exactly when a <bijection> pairs their members. For a <finite set>, cardinality is the ordinary number of <set elements>. In the presence of the <axiom of choice>, every <set> can be assigned the least <ordinal> in its <bijection> class as its <cardinal number>.
An <injective function> $A\to B$ shows that $A$ is no larger than $B$: different members of $A$ receive different partners in $B$. If there are <injections> in both directions, the <Cantor-Schröder-Bernstein theorem> supplies a <bijection>. Infinite <cardinality> records this pairing structure rather than a geometric dimension or density. That is why a line of <natural numbers> and a grid of natural-number pairs can have the same size, while an interval of <real numbers> has larger <cardinality>.
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