Definition:Equivalence Relation/Definition 2
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Definition
Let $\RR \subseteq S \times S$ be a relation on a set $S$.
$\RR$ is an equivalence relation if and only if:
- $\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$
where:
- $\Delta_S$ denotes the diagonal relation on $S$
- $\RR^{-1}$ denotes the inverse relation
- $\circ$ denotes composition of relations
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
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations