Injective function: Difference between revisions
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In [[mathematics]], an '''injective function''' or '''one-to-one function''' or '''injection''' is a [[function (mathematics)|function]] which has different output values on different input values: ''f'' is injective if <math>x_1 \neq x_2</math> implies that <math>f(x_1) \neq f(x_2)</math>. | In [[mathematics]], an '''injective function''' or '''one-to-one function''' or '''injection''' is a [[function (mathematics)|function]] which has different output values on different input values: ''f'' is injective if <math>x_1 \neq x_2</math> implies that <math>f(x_1) \neq f(x_2)</math>. | ||
An injective function ''f'' has a well-defined partial inverse <math>f^{-1}</math>. 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>. If ''f'' is injective then this ''x'' is unique and we can define <math>f^{-1}(y)</math> to be this unique value. We have <math>f^{-1}(f(x)) = x</math> for all ''x'' in the domain. | An injective function ''f'' has a well-defined partial inverse <math>f^{-1}</math>. 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>. If ''f'' is injective then this ''x'' is unique and we can define <math>f^{-1}(y)</math> to be this unique value. We have <math>f^{-1}(f(x)) = x</math> for all ''x'' in the domain. | ||
A strictly [[monotonic function]] is injective, since in this case <math>x_1 < x_2</math> implies that <math>f(x_1) < f(x_2)</math>. | |||
==See also== | ==See also== | ||
* [[Bijective function]] | * [[Bijective function]] | ||
* [[Surjective function]] | * [[Surjective function]] | ||
Latest revision as of 12:23, 13 November 2008
In mathematics, an injective function or one-to-one function or injection is a function which has different output values on different input values: f is injective if implies that .
An injective function f has a well-defined partial inverse . If y is an element of the image set of f, then there is at least one input x such that . If f is injective then this x is unique and we can define to be this unique value. We have for all x in the domain.
A strictly monotonic function is injective, since in this case implies that .
