Surjective function: Difference between revisions
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In [[mathematics]], a '''surjective function''' or '''onto function''' or '''surjection''' is a [[function (mathematics)|function]] for which every possible output value occurs for one or more input values: that is, its image is the whole of its codomain. | In [[mathematics]], a '''surjective function''' or '''onto function''' or '''surjection''' is a [[function (mathematics)|function]] for which every possible output value occurs for one or more input values: that is, its image is the whole of its codomain. | ||
An surjective function ''f'' has an inverse <math>f^{-1}</math> (this requires us to assume the [[ | An surjective function ''f'' has an inverse <math>f^{-1}</math> (this requires us to assume the [[Axiom of Choice]]). If ''y'' is an element of the image set of ''f'', then there is at least one input ''x'' such that <math>f(x) = y</math>. We define <math>f^{-1}(y)</math> to be one of these ''x'' values. We have <math>f(f^{-1}(y) = y</math> for all ''y'' in the codomain. | ||
==See also== | ==See also== | ||
* [[Bijective function]] | * [[Bijective function]] | ||
* [[Injective function]] | * [[Injective function]] | ||
Revision as of 14:38, 12 November 2008
In mathematics, a surjective function or onto function or surjection is a function for which every possible output value occurs for one or more input values: that is, its image is the whole of its codomain.
An surjective function f has an inverse (this requires us to assume the Axiom of Choice). If y is an element of the image set of f, then there is at least one input x such that . We define to be one of these x values. We have for all y in the codomain.
