**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: 2018-12-15 19:38:42 | 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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[573] **Schmidt Gunther, StrÃ¶hlein Thomas**: “Relationen und Graphen”, Springer-Verlag, 1989

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