Namespaces
Variants
Actions

Difference between revisions of "Surjection"

From Encyclopedia of Mathematics
Jump to: navigation, search
(more precise)
(expanded)
Line 1: Line 1:
 +
\def\Id {\mathop{\rm Id}}
  
 
A function (or mapping) is called '''surjective''' if the image of its domain coincides with its range.
 
A function (or mapping) is called '''surjective''' if the image of its domain coincides with its range.
Line 4: Line 5:
 
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  
 
: a '''surjective function''' or a '''surjection''' or a '''function onto''' A  
 
: a '''surjective function''' or a '''surjection''' or a '''function onto''' A  
iff
+
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 .
 
: f(A)=B , i.e., for each b \in B there is an a \in A such that f(a)=b .
A function that is both surjective and [[injective]] is called [[bijective]]  
+
 
 +
==== 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|Axiom of Choice]].
 +
 
 +
==== Related notions ====
 +
 
 +
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]]).
 
(or, if domain and range coincide, in some contexts, a [[permutation of a set|permutation]]).
 +
 +
An surjective [[homomorphism]] is called epimorphism.
 +
 +
Surjective mappings that are compatible with the underlying structure are often called [[projection]]s.

Revision as of 13:53, 18 February 2012

\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=21022
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article