Injective function: Difference between revisions

Revision as of 17:16, 25 June 2022

Revision as of 17:57, 25 June 2022

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== Injections can be undone ==

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>

:<math>g(f(x)) = x</math> (<math>f</math> can be undone by <math>g</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 non-empty ___domain has a left inverse <math>g,</math> which can be defined by fixing an element <math>a</math> in the ___domain of <math>f</math> so that <math>g(x)</math> equals the unique pre-image of <math>x</math> under <math>f</math> if it exists and <math>g(x) = 1</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}.}}