Definition:Equivalence Relation/Definition 1

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Let $\mathcal R$ be a relation on a set $S$.

Let $\mathcal R$ be:

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

Then $\mathcal R$ is an equivalence relation on $S$.

Also known as

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

Also denoted as

When discussing equivalence relations, various notations are used for $\left({x, y}\right) \in \mathcal R$.

Examples are:

$x \equiv \map y {\mathcal R}$
$x \equiv y \pmod {\mathcal R}$
$x \sim y$

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.