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equivalence relation
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(Definition)
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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  and take a positive integer $m$ . Then $m$ induces an equivalence relation by $a\sim b$ when $m$ divides $b-a$ (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 $ab^{-1}\in H$ . That defines an equivalence relation. Here equivalence classes are called cosets.
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"equivalence relation" is owned by CWoo. [ full author list (2) | owner history (2) ]
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Cross-references: cosets, subgroup, group, remainder, divides, positive, integers, unit, quotient space, conversely, partition, induces, relation
There are 218 references to this entry.
This is version 10 of equivalence relation, born on 2001-10-18, modified 2008-11-02.
Object id is 349, canonical name is EquivalenceRelation.
Accessed 43987 times total.
Classification:
| AMS MSC: | 06-00 (Order, lattices, ordered algebraic structures :: General reference works ) |
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Pending Errata and Addenda
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