Injective function: Difference between revisions

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Revision as of 05:13, 12 January 2023

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Functions with [[Inverse function#Left and right inverses|left inverses]] are always injections. That is, given <math>f : X \to Y,</math> if there is a function <math>g : Y \to X</math> such that for every <math>x \in X</math>, <math>g(f(x)) = x</math>, then <math>f</math> is injective. In this case, <math>g</math> is called a [[Retract (category theory)|retraction]] of <math>f.</math> Conversely, <math>f</math> is called a [[Retract (category theory)|section]] of <math>g.</math>

Conversely, every injection <math>f</math> with a non-empty ___domain has a left inverse <math>g,</math>. whichIt can be defined by fixingchoosing an element <math>a</math> in the ___domain of <math>f</math> soand thatsetting <math>g(xy)</math> equalsto the unique pre-imageelement of <math>x</math>the underpre-image <math>f^{-1}[y]</math> (if it existsis andnon-empty) or to <math>g(x) = 1a</math> (otherwise).{{refn|Unlike the corresponding statement that every surjective function has a right inverse, this does not require the [[axiom of choice]], as the existence of <math>a</math> is implied by the non-emptiness of the ___domain. However, this statement may fail in less conventional mathematics such as [[constructive mathematics]]. In constructive mathematics, the inclusion <math>\{ 0, 1 \} \to \R</math> of the two-element set in the reals cannot have a left inverse, as it would violate [[Indecomposability (constructive mathematics)|indecomposability]], by giving a [[Retract (category theory)|retraction]] of the real line to the set {0,1}.}}

The left inverse <math>g</math> is not necessarily an [[Inverse function|inverse]] of <math>f,</math> because the composition in the other order, <math>f \circ g,</math> may differ from the identity on <math>Y.</math> In other words, an injective function can be "reversed" by a left inverse, but is not necessarily [[Inverse function|invertible]], which requires that the function is bijective.