equivalence relation
An equivalence relation^{} $\sim $ on a set $S$ is a relation^{} that is:
 Reflexive^{}.

$a\sim a$ for all $a\in S$.
 Symmetric.

Whenever $a\sim b$, then $b\sim a$.
 Transitive^{}.

If $a\sim b$ and $b\sim c$ then $a\sim c$.
If $a$ and $b$ are related this way we say that they are equivalent^{} under $\sim $. If $a\in S$, then the set of all elements of $S$ that are equivalent to $a$ is called the equivalence class^{} of $a$. The set of all equivalence classes under $\sim $ is written $S/\sim $.
An equivalence relation on a set induces a partition on it. Conversely, any partition induces an equivalence relation. Equivalence relations are important, because often the set $S$ can be ’transformed’ into another set (quotient space^{}) by considering each equivalence class as a single unit.
Two examples of equivalence relations:
1. Consider the set of integers $\mathbb{Z}$ and take a positive integer $m$. Then $m$ induces an equivalence relation by $a\sim b$ when $m$ divides $ba$ (that is, $a$ and $b$ leave the same remainder when divided by $m$).
2. Take a group $(G,\cdot )$ and a subgroup^{} $H$. Define $a\sim b$ whenever $a{b}^{1}\in H$. That defines an equivalence relation. Here equivalence classes are called cosets.
Title  equivalence relation 
Canonical name  EquivalenceRelation 
Date of creation  20130322 11:48:27 
Last modified on  20130322 11:48:27 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  15 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 0600 
Classification  msc 03D20 
Related topic  QuotientGroup 
Related topic  EquivalenceClass 
Related topic  Equivalent 
Related topic  EquivalenceRelation 
Related topic  Partition 
Related topic  MathbbZ_n 
Defines  equivalent 
Defines  equivalence class 