# equivalence class

Let $S$ be a set with an equivalence relation  $\sim$. An 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