Difference between revisions of "Injection"
From Maths
m |
m |
||
| Line 1: | Line 1: | ||
An injective function is 1:1, but not nessasarally [[Surjection|onto]]. | An injective function is 1:1, but not nessasarally [[Surjection|onto]]. | ||
==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: | + | 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: | ||
| Line 7: | Line 7: | ||
==Notes== | ==Notes== | ||
| − | The cardinality of the inverse of an element <math>y\in Y</math> may be no more than 1 | + | ===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> |
| + | ** 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. | ||
| + | {{Todo|Investigate page 6 in Probability and Stochastics, it uses ''into'' and see if injectivity matters}} | ||
| + | ===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== | ==See also== | ||
Revision as of 18:09, 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]
- 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.
TODO: Investigate page 6 in Probability and Stochastics, it uses into and see if injectivity matters
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 Analysis: Part 1 - Elements - Krzysztof Maurin
- ↑ http://mathforum.org/library/drmath/view/52454.html
