# Surjection

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

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

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 if there is a right-inverse function $g$ with $ f \circ g = \Id_B$ (cf. Section of a mapping). 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*.

**How to Cite This Entry:**

Surjection.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Surjection&oldid=35689