equivalence class

Let S be a set with an equivalence relationMathworldPlanetmath . An equivalence classMathworldPlanetmath of S under is a subset TS such that

  • If xT and yS, then xy if and only if yT

  • If S is nonempty, then T is nonempty

For xS, the equivalence class containing x is often denoted by [x], so that


The set of all equivalence classes of S under is defined to be the set of all subsets of S which are equivalence classes of S under , and is denoted by S/. The map x[x] is sometimes referred to as the .

For any equivalence relation , the set of all equivalence classes of S under 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 EquivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath
Related topic Partition