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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:

• 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 partition the set $V$ into distinct, 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.

| | | | | created: 2018-12-15 23:06:21 | modified: 2018-12-15 23:56:58 | by: bookofproofs | references: [573], [577]

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