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equivalence relation
An equivalence relation $\sim$ on a set $S$ is a relation that is:
 Reflexive.

$a\sim a$ for all $a\in S$.
 Symmetric.

Whenever $a\sim b$, then $b\sim a$.
 Transitive.

If $a\sim b$ and $b\sim c$ then $a\sim c$.
If $a$ and $b$ are related this way we say that they are equivalent under $\sim$. If $a\in S$, then the set of all elements of $S$ that are equivalent to $a$ is called the equivalence class of $a$. The set of all equivalence classes under $\sim$ is written $S/\sim$.
An equivalence relation on a set induces a partition on it. Conversely, any partition induces an equivalence relation. Equivalence relations are important, because often the set $S$ can be ’transformed’ into another set (quotient space) by considering each equivalence class as a single unit.
Two examples of equivalence relations:
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integers modulo n
For example, the integers modulo n are partitioned into n equivalence classes, by the relation a R b iff remainder(a/n)=remainder(b/n).
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