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Difference between revisions of "Surjection"

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(→‎Equivalent conditions: revised formulation for right inverse (avoiding f^{-1} because it is not unique))
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$ \def\Id {\mathop{\rm Id}} $
 
$ \def\Id {\mathop{\rm Id}} $
  
A function (or mapping) is called '''surjective''' if the image of its domain coincides with its range.
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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  
 
In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is  
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$ f(f^{-1}(S)) = S $ for all subsets $S$ of the range $B$.
 
$ 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 $B$,  
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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 $.   
the condition $ g \circ f = h \circ f $  implies $ g=h $.   
 
  
A function $f$ is surjective there is a right-inverse function $g$ with $ f \circ g = \Id_A$.
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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|Axiom of Choice]].
The converse statement is equivalent to the [[axiom of choice|Axiom of Choice]].
 
  
 
==== Related notions ====
 
==== Related notions ====
  
A function that is both surjective and [[Injection|injective]] is called [[Bijection|bijective]]  
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A function that is both surjective and [[Injection|injective]] is called ''[[Bijection|bijective]]''
(or, if domain and range coincide, in some contexts, a [[permutation of a set|permutation]]).
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(or, if domain and range coincide, in some contexts, a ''[[permutation of a set|permutation]])''.
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An surjective [[homomorphism]] is called ''epimorphism''.  
  
An surjective [[homomorphism]] is called epimorphism.  
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Surjective mappings that are compatible with the underlying structure are often called ''[[projection]]s''.
  
Surjective mappings that are compatible with the underlying structure are often called [[projection]]s.
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[[Category:Set theory]]

Latest revision as of 21:19, 18 December 2014


$ \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=21201
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article