Editing Inverse function (section)

====Left inverses====
If {{math|''f'': ''X'' → ''Y''}}, a '''left inverse''' for {{mvar|f}} (or ''[[retract (category theory)|retraction]]'' of {{mvar|f}} ) is a function {{math| ''g'': ''Y'' → ''X''}} such that composing {{mvar|f}} with {{mvar|g}} from the left gives the identity function<ref>{{cite book|last1=Dummit|last2=Foote|title=Abstract Algebra}}</ref> <math display="block">g \circ f = \operatorname{id}_X\text{.}</math>  That is, the function {{mvar|g}} satisfies the rule
: If {{math|''f''(''x''){{=}}''y''}}, then {{math|''g''(''y''){{=}}''x''}}.

The function {{mvar|g}} must equal the inverse of {{mvar|f}} on the image of {{mvar|f}}, but may take any values for elements of {{mvar|Y}} not in the image.

A function {{mvar|f}} with nonempty domain is injective if and only if it has a left inverse.<ref>{{cite book|last=Mac&nbsp;Lane|first=Saunders|title=Categories for the Working Mathematician}}</ref>  An elementary proof runs as follows:
* If {{mvar|g}} is the left inverse of {{mvar|f}}, and {{math|1=''f''(''x'') = ''f''(''y'')}}, then {{math|1=''g''(''f''(''x'')) = ''g''(''f''(''y'')) = ''x'' = ''y''}}.
* <p>If nonempty {{math|''f'': ''X'' → ''Y''}} is injective, construct a left inverse {{math|''g'': ''Y'' → ''X''}} as follows: for all {{math|''y'' ∈ ''Y''}}, if {{mvar|y}} is in the image of {{mvar|f}}, then there exists {{math|''x'' ∈ ''X''}} such that {{math|1=''f''(''x'') = ''y''}}.  Let {{math|1=''g''(''y'') = ''x''}}; this definition is unique because {{mvar|f}} is injective.  Otherwise, let {{math|''g''(''y'')}} be an arbitrary element of {{mvar|X}}.</p><p>For all {{math|''x'' ∈ ''X''}}, {{math|''f''(''x'')}} is in the image of {{mvar|f}}.  By construction, {{math|1=''g''(''f''(''x'')) = ''x''}}, the condition for a left inverse.</p>

In classical mathematics, every injective function {{mvar|f}} with a nonempty domain necessarily has a left inverse; however, this may fail in [[constructive mathematics]]. For instance, a left inverse of the [[Inclusion map|inclusion]] {{math|{0,1} → '''R'''}} of the two-element set in the reals violates [[indecomposability (constructive mathematics)|indecomposability]] by giving a [[Retract (category theory)|retraction]] of the real line to the set {{math|{0,1{{)}}}}.<ref>{{cite journal|last=Fraenkel|title=Abstract Set Theory|journal=Nature |year=1954 |volume=173 |issue=4412 |page=967 |doi=10.1038/173967a0 |bibcode=1954Natur.173..967C |s2cid=7735523 |doi-access=free }}</ref>