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Surjection

From Encyclopedia of Mathematics
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$ \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.

How to Cite This Entry:
Surjection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surjection&oldid=21173
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article