Difference between revisions of "Injection"

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{{Requires work|grade=A*
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|msg=This needs to be modified (in tandem with [[Surjection]]) to:
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# 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)
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# be linked to [[cardinality of sets]] and that Cantor theorem. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:50, 8 May 2018 (UTC)}}
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An injective function is 1:1, but not nessasarally [[Surjection|onto]].
 
An injective function is 1:1, but not nessasarally [[Surjection|onto]].
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__TOC__
 
==Definition==
 
==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:
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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>
 
* <math>\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2]</math>
 
Or equivalently:
 
Or equivalently:
* <math>\forall x_1,x_2\in X[x_1\ne x_2\implies f(x_1)=f(x_2)]</math> (the [[Contrapositive|contrapositive]] of the above)
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* <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)
 
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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.
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==Statements==
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* [[Every injection yields a bijection onto its image]]
 
==Notes==
 
==Notes==
The cardinality of the inverse of an element <math>y\in Y</math> may be no more than 1; that is 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)
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===Terminology===
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*An injective function is sometimes called an ''embedding''<ref name="API"/>
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*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.
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** 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)
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** '''I do not like using the word ''into'' but do like ''onto'' - I say:'''
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**: ''"But {{M|f}} maps {{M|A}} onto {{M|B}} so...."''
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**: ''"But {{M|f}} is an injection so...."''
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**: ''"As {{M|f}} is a bijection..."''
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** 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>
  
{{Todo|Find reference - should be easy!}}
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===Properties===
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* The cardinality of the inverse of an element <math>y\in Y</math> may be no more than 1
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** Note this means it may be zero
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**: 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==
 
==See also==
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==References==
 
==References==
 
<references/>
 
<references/>
 
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{{Function terminology navbox|plain}}
 
{{Definition|Set Theory}}
 
{{Definition|Set Theory}}

Latest revision as of 21:50, 8 May 2018

Grade: A*
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The message provided is:
This needs to be modified (in tandem with Surjection) to:
  1. allow surjection/injection/bijection to be seen through the lens of Category Theory. Alec (talk) 21:50, 8 May 2018 (UTC)
  2. be linked to cardinality of sets and that Cantor theorem. Alec (talk) 21:50, 8 May 2018 (UTC)

An injective function is 1:1, but not nessasarally onto.

Definition

For a 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[1]:

  • [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 of the above)

Sometimes an injection is denoted [ilmath]\rightarrowtail[/ilmath][2] (and a surjection [ilmath]\twoheadrightarrow[/ilmath] 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

Notes

Terminology

  • An injective function is sometimes called an embedding[1]
  • Just as surjections are called 'onto' an injection may be called 'into'[3] 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 [ilmath]f[/ilmath] maps [ilmath]A[/ilmath] onto [ilmath]B[/ilmath] so...."
      "But [ilmath]f[/ilmath] is an injection so...."
      "As [ilmath]f[/ilmath] is a bijection..."
    • I see into used rarely to mean injection, and in fact any function [ilmath]f:X\rightarrow Y[/ilmath] being read as [ilmath]f[/ilmath] takes [ilmath]X[/ilmath] into [ilmath]Y[/ilmath] without meaning injection[1][4]

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 [ilmath]f^{-1}(y)=x[/ilmath])

See also

References

  1. 1.0 1.1 1.2 Analysis: Part 1 - Elements - Krzysztof Maurin
  2. Notes On Set Theory - Second Edition - Yiannis Moschovakis
  3. http://mathforum.org/library/drmath/view/52454.html
  4. Real and Abstract Analysis - Edwin Hewitt and Karl Stromberg