equivalence class
Let S be a set with an equivalence relation ∼. An equivalence class
of S under ∼ is a subset T⊂S such that
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•
If x∈T and y∈S, then x∼y if and only if y∈T
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•
If S is nonempty, then T is nonempty
For x∈S, the equivalence class containing x is often denoted by [x], so that
[x]:= |
The set of all equivalence classes of under is defined to be the set of all subsets of which are equivalence classes of under , and is denoted by . The map is sometimes referred to as the .
For any equivalence relation , the set of all equivalence classes of under is a partition of , and this correspondence is a bijection between the set of equivalence relations on and the set of partitions of (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 |