Equinumerous sets

Definition

Let $A$ and $B$ be sets. The sets $A$ and $B$ are equinumerous, written $A \approx B$, when there is a bijective function $f \colon A \to B$.

Remarks

Equinumerosity is reflexive, symmetric and transitive. The identity, the inverse and the composition of bijections give the three properties. The set $\mathbb{N}$ is equinumerous with the set of even natural numbers, by $n \mapsto 2n$. So an infinite set can be equinumerous with a proper subset. Other names are "equipotent" and "have the same cardinality".

Explanations

L1 human

You do not need to count to compare sizes. If each chair in a room has one person and each person has one chair, there are as many people as chairs. Equinumerosity applies the same idea to infinite sets.