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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.

| | | | | created: 2014-04-02 23:32:50 | modified: 2019-07-28 19:15:15 | by: bookofproofs | references: [573]

1.Motivation: Significance of Equivalence Relations

2.Example: Examples of Equivalence Relations

3.Proposition: The Equality of Sets Is an Equivalence Relation

4.Definition: Equivalence Class

5.Definition: Quotient Set, Partition

6.Definition: Complete System of Representatives

7.Definition: Canonical Projection

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Bibliography (further reading)

[573] Schmidt Gunther, Ströhlein Thomas: “Relationen und Graphen”, Springer-Verlag, 1989