Difference between revisions of "Equivalence class"

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==Equivalence classes are either the same or disjoint==
 
==Equivalence classes are either the same or disjoint==
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Suppose there were two equivalence classes <math>[a]</math> and <math>[b]</math>. We can write the members of each class as <math>[a_n]</math> and <math>[b_n]</math>.
 
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Suppose the two classes were both nonidentical and nondisjoint. Then there exists <math>[a_1] \sim [b_1]</math> and <math>[a_2] \nsim [b_2]</math>. However, <math>[a_1] \sim [a_2]</math> and <math>[b_1] \sim [b_2]</math>, thus <math>[a_2] \sim [b_2]</math>, a contradiction. Therefore the classes must be either identical or disjoint.
 
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This is the motivation for how [[Coset|cosets]] partition groups.
 
This is the motivation for how [[Coset|cosets]] partition groups.
  

Latest revision as of 20:00, 14 November 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

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