Difference between revisions of "Surjection"
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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. | A function (or mapping) is called '''surjective''' if the image of its domain coincides with its range. | ||
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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''' $A$ | ||
| − | + | 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.
Surjection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surjection&oldid=21173