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. | ||
Revision as of 19:25, 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
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].
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.
This is the motivation for how cosets partition groups.
References
- ↑ Functional Analysis - George Bachman and Lawrence Narici
TODO: Add proofs and whatnot
