Bijective function

Definition

Let $f \colon A \to B$ be a function. The function $f$ is bijective when it is injective and surjective. Equivalently, for each $b \in B$, there is exactly one $a \in A$ with $f(a) = b$.

Remarks

Other names are bijection and one-to-one correspondence. The function $x \mapsto 2x + 1$ on $\mathbb{R}$ is bijective. A function is bijective if and only if it has an inverse function. See bijective iff it has a two-sided inverse. Two finite sets have the same number of elements exactly when there is a bijection between them.

Explanations

L1 human

A bijection pairs the elements of $A$ and the elements of $B$ perfectly. Each element of $A$ has one partner in $B$, and each element of $B$ has one partner in $A$.