Surjection

$\def\Id {\mathop{\rm Id}}$

A function (or mapping) is called surjective if the image of its domain coincides with its range.

In other words, a function $f : A \to B$ from a set $A$ to a set $B$ is

a surjective function or a surjection or a function onto $A$

if and only if

$f(A)=B$, i.e., for each $b \in B$ there is an $a \in A$ such that $f(a)=b$.

Equivalent conditions

A function $f$ is surjective if and only if $f(f^{-1}(S)) = S$ for all subsets $S$ of the range $B$.

A function $f$ is surjective if and only if, for every pair of functions $g,h$ defined on $B$, the condition $g \circ f = h \circ f$ implies $g=h$.

A function $f$ is surjective it it has a right inverse $f^{-1}$ with $f \circ f^{-1} = \Id_A$. The converse statement is equivalent to the Axiom of Choice.

Related notions

A function that is both surjective and injective is called bijective (or, if domain and range coincide, in some contexts, a permutation).

An surjective homomorphism is called epimorphism.

Surjective mappings that are compatible with the underlying structure are often called projections.