equivalence class
Let $S$ be a set with an equivalence relation^{} $\sim $. An equivalence class^{} of $S$ under $\sim $ is a subset $T\subset S$ such that

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If $x\in T$ and $y\in S$, then $x\sim y$ if and only if $y\in T$

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If $S$ is nonempty, then $T$ is nonempty
For $x\in S$, the equivalence class containing $x$ is often denoted by $[x]$, so that
$$[x]:=\{y\in S\mid x\sim y\}.$$ 
The set of all equivalence classes of $S$ under $\sim $ is defined to be the set of all subsets of $S$ which are equivalence classes of $S$ under $\sim $, and is denoted by $S/\sim $. The map $x\mapsto [x]$ is sometimes referred to as the .
For any equivalence relation $\sim $, the set of all equivalence classes of $S$ under $\sim $ is a partition of $S$, and this correspondence is a bijection between the set of equivalence relations on $S$ and the set of partitions of $S$ (consisting of nonempty sets).
Title  equivalence class 
Canonical name  EquivalenceClass 
Date of creation  20130322 11:52:30 
Last modified on  20130322 11:52:30 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  10 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 03E20 
Classification  msc 93D05 
Classification  msc 03B52 
Classification  msc 93C42 
Related topic  EquivalenceRelation 
Related topic  Equivalent^{} 
Related topic  Partition 