Namespaces
Variants
Actions

Difference between revisions of "Surjection"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX.)
(rewritten)
Line 1: Line 1:
−
''surjective mapping, from a set $A$ onto a set $B$''
 
  
−
A mapping $f$ such that $f(A)=B$, i.e. such that for each $b\in B$ there is an $a\in A$ with $f(a)=b$. As well as saying  "$f$ is surjective" , one can also say  "$f$ is a mapping from $A$ onto $B$" .
+
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''' $A$
−
====Comments====
+
iff
−
See also [[Injection|Injection]]; [[Bijection|Bijection]]; [[Permutation of a set|Permutation of a set]].
+
: $ 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]]  
 +
(or, in some contexts, a [[permutation of a set|permutation]]).

Revision as of 01:36, 14 February 2012

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 $A$

iff

$ 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 (or, in some contexts, a permutation).

How to Cite This Entry:
Surjection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surjection&oldid=21018
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Surjection&oldid=21019"