Difference between revisions of "Equivalence class"

Latest revision as of 20:00, 14 November 2015

Given an Equivalence relation $\sim$ the equivalence class of $a$ is denoted as follows:

$[a]=\{b|a\sim b\}$

An equivalence class may be denoted by $[a]$ where $a$ is the representative of it. There is an alternative representation:

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 $[a]_{\sim_1}$ for a class in $\sim_1$ and $[f]_{\sim_2}$ for $\sim_2$

An equivalence relation is a partition

This is the motivation for how cosets partition groups.

TODO: Add proofs and whatnot

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 ==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 [[Coset|cosets]] partition groups.