Difference between revisions of "Equivalence class"
From Maths
(Created page with "==Definition== Given an Equivalence relation {{M|\equiv}} the equivalence class of {{M|a}} is denoted as follows: <math>[a]=\{b|a\equiv b\}</math> ==Equivalence relation...") |
|||
| (3 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
==Definition== | ==Definition== | ||
| − | Given an [[Equivalence relation]] {{M|\ | + | Given an [[Equivalence relation]] {{M|\sim}} the equivalence class of {{M|a}} is denoted as follows: |
| − | <math>[a]=\{b|a\ | + | <math>[a]=\{b|a\sim b\}</math> |
| + | ==Notations== | ||
| + | An equivalence class may be denoted by {{M|[a]}} where {{M|a}} is the ''representative'' of it. There is an alternative representation: | ||
| + | * {{M|\hat{a} }}, where again {{M|a}} is the representative of the class.<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref> | ||
| + | I quite like the hat notation, however I recommend one ''avoids'' using it when there are multiple [[Equivalence relations]] at play. | ||
| + | If there are multiple ones, then we can write for example {{M|[a]_{\sim_1} }} for a class in {{M|\sim_1}} and {{M|[f]_{\sim_2} }} for {{M|\sim_2}} | ||
==Equivalence relations partition sets== | ==Equivalence relations partition sets== | ||
An equivalence relation is a partition | An equivalence relation is a partition | ||
| Line 9: | Line 14: | ||
==Equivalence classes are either the same or disjoint== | ==Equivalence classes are either the same or disjoint== | ||
This is the motivation for how [[Coset|cosets]] partition groups. | This is the motivation for how [[Coset|cosets]] partition groups. | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Todo|Add proofs and whatnot}} | {{Todo|Add proofs and whatnot}} | ||
{{Definition|Set Theory|Abstract Algebra}} | {{Definition|Set Theory|Abstract Algebra}} | ||
Latest revision as of 20:00, 14 November 2015
Contents
Definition
Given an Equivalence relation [ilmath]\sim[/ilmath] the equivalence class of [ilmath]a[/ilmath] is denoted as follows:
[math][a]=\{b|a\sim b\}[/math]
Notations
An equivalence class may be denoted by [ilmath][a][/ilmath] where [ilmath]a[/ilmath] is the representative of it. There is an alternative representation:
- [ilmath]\hat{a} [/ilmath], where again [ilmath]a[/ilmath] is the representative of the class.[1]
I quite like the hat notation, however I recommend one avoids using it when there are multiple Equivalence relations at play.
If there are multiple ones, then we can write for example [ilmath][a]_{\sim_1} [/ilmath] for a class in [ilmath]\sim_1[/ilmath] and [ilmath][f]_{\sim_2} [/ilmath] for [ilmath]\sim_2[/ilmath]
Equivalence relations partition sets
An equivalence relation is a partition
Equivalence classes are either the same or disjoint
This is the motivation for how cosets partition groups.
References
- ↑ Functional Analysis - George Bachman and Lawrence Narici
TODO: Add proofs and whatnot
