Definition · applies to sets · also called $\lvert A \rvert$, $\#A$

Elementary

The cardinality of a set is how many elements it has.

Caveat. For infinite sets, “how many” needs a new idea: two sets have the same cardinality if their elements can be paired up one-to-one.
High school

The cardinality of a finite set is its number of elements.

Graduate (set theory)

Sets $A$ and $B$ have the same cardinality if there is a bijection between them. This extends counting to infinite sets.

Properties

  • Infinite sets can have different cardinalities: $\lvert \mathbb{N} \rvert < \lvert \reals \rvert$.
  • Every set has smaller cardinality than its power set (Cantor’s theorem).

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