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

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

• 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	2013-03-22 11:52:30
Last modified on	2013-03-22 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