Difference between revisions of "Surjection"
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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. | + | 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 $B$, | + | 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 = \ | + | 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]] | + | 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]])''. |
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| + | An surjective [[homomorphism]] is called ''epimorphism''. | ||
| − | + | Surjective mappings that are compatible with the underlying structure are often called ''[[projection]]s''. | |
| − | + | [[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.
Surjection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surjection&oldid=21201