The unique properties of any equivalence relation $R\subseteq V\times V$ have important conseqences for the structure of its equivalence classes: * Every equivalence $[a]$ is non-empty for all $a\in V.$ This follows from $R$ being reflexive. * If $c\in [a]$ and $c\in [b]$ then $a\sim c\sim b$. Therefore $[a]=[b]$. This follows from $R$ being transitive. * Of course, from $R$ being symmetric, we also have $b\sim c\sim a$. In fact, the two equivalence classes are equal $[a]=[b]$ if and only if $a\sim b.$ In other words, $[a]$ and $b$ are equal if $a\sim b$, otherwise they are disjoint $[a]\cap [b]=\emptyset$.

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:

• $V$ modulo $R$, or
• partition of the set $V$ under $R$.

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

Axioms: 1
Definitions: 2 3 4 5 6
Explanations: 7
Parts: 8
Proofs: 9 10
Propositions: 11 12 13 14 15
Solutions: 16

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### References

#### Bibliography

1. Schmidt G., Ströhlein T.: "Relationen und Graphen", Springer-Verlag, 1989
2. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001