Difference between revisions of "Equivalence class"

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<math>[a]=\{b|a\sim b\}</math>
 
<math>[a]=\{b|a\sim b\}</math>
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==Notations==
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An equivalence class may be denoted by {{M|[a]}} where {{M|a}} is the ''representative'' of it. There is an alternative representation:
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* {{M|\hat{a} }}, where again {{M|a}} is the representative of the class.<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>
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I quite like the hat notation, however I recommend one ''avoids'' using it when there are multiple [[Equivalence relations]] at play.
  
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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
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==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.
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==References==
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<references/>
  
 
{{Todo|Add proofs and whatnot}}
 
{{Todo|Add proofs and whatnot}}
 
{{Definition|Set Theory|Abstract Algebra}}
 
{{Definition|Set Theory|Abstract Algebra}}

Revision as of 19:48, 10 July 2015

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

  1. Functional Analysis - George Bachman and Lawrence Narici



TODO: Add proofs and whatnot