Relation set

Equivalence relation

An equivalence relation is any relation with the properties of

1. reflexivity
2. symmetry
3. transitivity

Quintessential examples include and isomorphic objects. A structure-preserving equivalence relation is called a Congruence relation, which precedes the notion of an Algebraic quotient.

Equivalence relations may be induced by a function: Given , then defines an equivalence relation on the set for any equivalence relation on the set .

Equivalence class

Every equivalence relation has a corresponding Partition of equivalence classes and vice versa.¹ An equivalence class for under is defined as

And has the following properties

• for any ,
• if and only if
• if and only if

The set of equivalence classes is called the Algebraic quotient.

Natural projection

Equivalence relations on a set are also characterised precisely by surjective functions called the natural projection whose fibres are equivalence classes. Then we say , with the natural isomorphism . If is a homomorphism then the induced equivalence relation is a congruence relation.

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Footnotes

1. 2017. Contemporary abstract algebra, p. 20 (Theorem 0.7) ↩