Difference between revisions of "Surjection"
(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 | ||
− | + | 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=21022