Definition:Equivalence Relation/Definition 2

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Definition

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


$\mathcal R$ is an equivalence relation if and only if:

$\Delta_S \cup \mathcal R^{-1} \cup \mathcal R \circ \mathcal R \subseteq \mathcal R$

where:

$\Delta_S$ denotes the diagonal relation on $S$
$\mathcal R^{-1}$ denotes the inverse relation
$\circ$ denotes composition of relations


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.


Sources