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Definition: Equivalence Relation

Let \(V\) be a set and let \(R\subseteq V\times V\) be a relation \(R\) is called an equivalence relation, if it is reflexive, symmetric and transitive. Elements of $V$ with $aRb$ are called equivalent. Other common notations are \(a\sim_R b\) or \(a\sim b\), if $R$ is known from the context.

Table of Contents

Motivations: 1 Examples: 1

1. Proposition: The Equality of Sets Is an Equivalence Relation
2. Definition: Equivalence Class
3. Definition: Quotient Set, Partition
4. Definition: Complete System of Representatives
5. Definition: Canonical Projection

Mentioned in:

Definitions: 1 2 3 4 5
Examples: 6
Explanations: 7
Lemmas: 8
Motivations: 9
Parts: 10 11
Proofs: 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Propositions: 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
Solutions: 44
Theorems: 45

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References

Bibliography

1. Schmidt G., Ströhlein T.: "Relationen und Graphen", Springer-Verlag, 1989