Difference between revisions of "Injection"
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===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:''' | ||
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**: ''"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 | ||
Revision as of 18:39, 28 August 2015
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)=f(x_2)][/math] (the contrapositive of the above)
Notes
Terminology
- An injective function is sometimes called an embedding[1]
- Just as surjections are called 'onto' an injection may be called 'into'[2] 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][3]
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
