Difference between revisions of "Injection"
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| + | |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)}} | ||
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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]]. | ||
| + | __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<ref name="API">Analysis: Part 1 - Elements - Krzysztof Maurin</ref>: | 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) | + | * <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== | ==Notes== | ||
===Terminology=== | ===Terminology=== | ||
*An injective function is sometimes called an ''embedding''<ref name="API"/> | *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> | + | *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) | ** 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:''' | ** '''I do not like using the word ''into'' but do like ''onto'' - I say:''' | ||
| Line 15: | Line 23: | ||
**: ''"But {{M|f}} is an injection so...."'' | **: ''"But {{M|f}} is an injection so...."'' | ||
**: ''"As {{M|f}} is a bijection..."'' | **: ''"As {{M|f}} is a bijection..."'' | ||
| − | ** I see ''into'' used rarely to mean injection | + | ** 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=== | ===Properties=== | ||
* The cardinality of the inverse of an element <math>y\in Y</math> may be no more than 1 | * The cardinality of the inverse of an element <math>y\in Y</math> may be no more than 1 | ||
| Line 29: | Line 37: | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | + | {{Function terminology navbox|plain}} | |
{{Definition|Set Theory}} | {{Definition|Set Theory}} | ||
Latest revision as of 21:50, 8 May 2018
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This needs to be modified (in tandem with Surjection) to:
- allow surjection/injection/bijection to be seen through the lens of Category Theory. Alec (talk) 21:50, 8 May 2018 (UTC)
- 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])
- Note this means it may be zero
See also
References
- ↑ 1.0 1.1 1.2 Analysis: Part 1 - Elements - Krzysztof Maurin
- ↑ Notes On Set Theory - Second Edition - Yiannis Moschovakis
- ↑ http://mathforum.org/library/drmath/view/52454.html
- ↑ Real and Abstract Analysis - Edwin Hewitt and Karl Stromberg
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