Cardinality

A cardinality is the size of a set, understood up to bijection: two sets $A$ and $B$ have the same cardinality if there exists a bijective function $f:A\to B$. The cardinality of $A$ is often denoted $|A|$.

Cardinality comparisons can be expressed using injections and surjections between sets. A set is countable when its cardinality is at most that of the natural numbers .

Examples:

• The empty set has cardinality $|\varnothing|=0$.
• If $A=\{1,2,3\}$, then $|A|=3$, and the power set $\mathcal{P}(A)$ has $|\mathcal{P}(A)|=8$.