View source for Injection

{{Requires work|grade=A*
|msg=This needs to be modified (in tandem with [[Surjection]]) to:
# allow surjection/injection/[[bijection]] to be seen through the lens of [[Category Theory]]. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:50, 8 May 2018 (UTC)
# be linked to [[cardinality of sets]] and that Cantor theorem. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:50, 8 May 2018 (UTC)}}

An injective function is 1:1, but not nessasarally [[Surjection|onto]].
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==Definition==
For a [[Function|function]] <math>f:X\rightarrow Y</math> every element of <math>X</math> is mapped to an element of <math>Y</math> and no two distinct things in <math>X</math> are mapped to the same thing in <math>Y</math>. That is<ref name="API">Analysis: Part 1 - Elements - Krzysztof Maurin</ref>:
* <math>\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2]</math>
Or equivalently:
* <math>\forall x_1,x_2\in X[x_1\ne x_2\implies f(x_1)\ne f(x_2)]</math> (the [[Contrapositive|contrapositive]] of the above)
Sometimes an injection is denoted {{M|\rightarrowtail}}{{rNOSTYM}} (and a [[surjection]] {{M|\twoheadrightarrow}} and a [[bijection]] is both of these combined (as if super-imposed on top of each other) - there is no LaTeX arrow for this however) - we do not use this convention.
==Statements==
* [[Every injection yields a bijection onto its image]]
==Notes==
===Terminology===
*An injective function is sometimes called an ''embedding''<ref name="API"/>
*Just as [[Surjection|surjections]] are called 'onto' an injection may be called 'into'<ref>http://mathforum.org/library/drmath/view/52454.html</ref> however this is rare and something I frown upon.
** This is French, from "throwing into" referring to the domain, not elements themselves (as any function takes an element ''into'' the codomain, it need not be one-to-one)
** '''I do not like using the word ''into'' but do like ''onto'' - I say:'''
**: ''"But {{M|f}} maps {{M|A}} onto {{M|B}} so...."''
**: ''"But {{M|f}} is an injection so...."''
**: ''"As {{M|f}} is a bijection..."''
** I see ''into'' used rarely to mean injection, and in fact any function {{M|f:X\rightarrow Y}} being read as {{M|f}} takes {{M|X}} into {{M|Y}} '''without''' meaning injection<ref name="API">Analysis: Part 1 - Elements - Krzysztof Maurin</ref><ref name="RAAA">Real and Abstract Analysis - Edwin Hewitt and Karl Stromberg</ref>

===Properties===
* The cardinality of the inverse of an element <math>y\in Y</math> may be no more than 1
** Note this means it may be zero
**: In contrast to a bijection where the cardinality is always 1 (and thus we take the singleton set <math>f^{-1}(y)=\{x\}</math> as the value it contains, writing {{M|1=f^{-1}(y)=x}})

==See also==
* [[Bijection]]
* [[Surjection]]
* [[Function]]

==References==
<references/>
{{Function terminology navbox|plain}}
{{Definition|Set Theory}}