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The unique properties of any equivalence relation $R\subseteq V\times V$ have important conseqences for the structure of its equivalence classes:

Thus, all equivalence classes are a partition of $V,$ because they are mutually disjoint, non-empty subsets: its equivalence classes. Therefore, every equivalence relation $R\subseteq V\times V$ generates another set, which we now want to define formally:

Definition: Quotient Set, Partition

Let $R\subseteq V\times V$ be an equivalence relation. The set of the corresponding equivalence classes $V/_{R}:=\{[a]|:a\in V\}$ is called the quotient set of $V$ by $R$.

The symbol $V/_{R}$ is sometimes also read:

The existence of the quotient set is ensured by the axiom of choice.

| | | | | created: 2018-12-15 23:06:21 | modified: 2019-09-08 11:16:48 | by: bookofproofs | references: [573], [577]

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Bibliography (further reading)

[577] Knauer Ulrich: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001

[573] Schmidt G., Ströhlein T.: “Relationen und Graphen”, Springer-Verlag, 1989