Cardinality

Two sets $A$ and $B$ are said to have the same cardinality if there exists a bijection $f\colon A\to B$. In that case one writes $|A|=|B|$.

For finite sets, $|A|=n$ means there is a bijection between $A$ and $\{1,2,\dots,n\}$. More generally, one compares cardinalities by the existence of injections and bijections (e.g. $|A|\le |B|$ if there is an injection $A\to B$).

A set is countable if its cardinality is at most that of $\mathbb{N}$.

Examples:

• $|\{a,b,c\}|=3$.
• $|\mathbb{Z}|=|\mathbb{N}|$ (there exists a bijection, though not an order-preserving one).
• If $A\subseteq B$ (see subset ) and both are finite, then $|A|\le |B|$.