equivalence relation

An equivalence relation^{👁 Mathworld
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on a set is a relation^{👁 Mathworld
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👁 Planetmath}
that is:

Reflexive^{👁 Mathworld
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.

for all .

Symmetric.

Whenever , then .

Transitive^{👁 Mathworld
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👁 Planetmath
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.

If and then .

If and are related this way we say that they are equivalent^{👁 Mathworld
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under . If , then the set of all elements of that are equivalent to is called the equivalence class^{👁 Mathworld
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of . The set of all equivalence classes under is written .

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 can be ’transformed’ into another set (quotient space^{👁 Mathworld
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) 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 . Then induces an equivalence relation by when divides (that is, and leave the same remainder when divided by ).

2. Take a group and a subgroup^{👁 Mathworld
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. Define whenever . That defines an equivalence relation. Here equivalence classes are called cosets.

Title	equivalence relation
Canonical name	EquivalenceRelation
Date of creation	2013-03-22 11:48:27
Last modified on	2013-03-22 11:48:27
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	15
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06-00
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