bijection

A bijective function (bijection) is a function between two sets such that every element of the codomain is the image of exactly one element of the domain, and every element is paired.

Formally,
a binary relation $f : X \to Y$ is a bijection if it is

left-total (total relation)
left-unique (injective relation)
right-total (surjective relation)
right-unique (functional relation)

A function $f : X \to Y$ is bijective if it is invertible; that is, if and only if
there exists a function $f^{- 1} : Y \to X$ such that composing the two functions in either direction produces an identity function:

$f^{- 1} (f (x)) = x$ for each $x \in X$
$f (f^{- 1} (y)) = y$ for each $y \in Y$