Definition:Equivalence Relation/Definition 1

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Definition

Let $\RR$ be a relation on a set $S$.


Let $\RR$ be:

$(1): \quad$ reflexive
$(2): \quad$ symmetric
$(3): \quad$ transitive

Then $\RR$ is an equivalence relation on $S$.


Also known as

An equivalence relation is frequently referred to just as an equivalence.

However, this usage is not recommended on $\mathsf{Pr} \infty \mathsf{fWiki}$, as it can obscure clarity.


Also denoted as

When discussing equivalence relations, various notations are used for $\tuple {x, y} \in \RR$.

Examples are:

$x \mathrel \RR y$
$x \equiv \map y \RR$
$x \equiv y \pmod \RR$

and so on.

Specialised equivalence relations generally have their own symbols, which can be defined as they are needed.

Such symbols include:

$\cong$, $\equiv$, $\sim$, $\simeq$, $\approx$


Also see

  • Results about equivalence relations can be found here.


Sources

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