# 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

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{I}$: Problem $\text{AA}$