Difference between revisions of "Injection"
From Maths
(Created page with "An injective function is 1:1, but not nessasarally onto. For <math>f:X\rightarrow Y</math> every element of <math>X</math> is mapped to an element of <math>Y</...") |
m (Adding "requires work" box about category theory and cardinality (set theory)) |
||
| (7 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | {{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)}} | ||
| − | For <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>. | + | An injective function is 1:1, but not nessasarally [[Surjection|onto]]. |
| + | __TOC__ | ||
| + | ==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]] | ||
| − | {{Definition}} | + | ==References== |
| + | <references/> | ||
| + | {{Function terminology navbox|plain}} | ||
| + | {{Definition|Set Theory}} | ||
Latest revision as of 21:50, 8 May 2018
Grade: A*
This page requires some work to be carried out
Some aspect of this page is incomplete and work is required to finish it
The message provided is:
The message provided is:
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.
Contents
[hide]Definition
For a function f:X→Y every element of X is mapped to an element of Y and no two distinct things in X are mapped to the same thing in Y. That is[1]:
- ∀x1,x2∈X[f(x1)=f(x2)⟹x1=x2]
Or equivalently:
- ∀x1,x2∈X[x1≠x2⟹f(x1)≠f(x2)] (the contrapositive of the above)
Sometimes an injection is denoted ↣[2] (and a surjection \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
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 f maps A onto B so...."
- "But f is an injection so...."
- "As f is a bijection..."
- I see into used rarely to mean injection, and in fact any function f:X\rightarrow Y being read as f takes X into Y without meaning injection[1][4]
Properties
- The cardinality of the inverse of an element y\in Y 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 f^{-1}(y)=\{x\} as the value it contains, writing f^{-1}(y)=x)
- Note this means it may be zero
See also
References
- ↑ Jump up to: 1.0 1.1 1.2 Analysis: Part 1 - Elements - Krzysztof Maurin
- Jump up ↑ Notes On Set Theory - Second Edition - Yiannis Moschovakis
- Jump up ↑ http://mathforum.org/library/drmath/view/52454.html
- Jump up ↑ Real and Abstract Analysis - Edwin Hewitt and Karl Stromberg
| |||||||||||
