# Definition:Equivalence Relation

## Definition

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

### Definition 1

Let $\RR$ be:

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

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

### Definition 2

$\RR$ 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.

## 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:

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

## Examples

### Same Age Relation

Let $P$ be the set of people.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { the age of$x$and$y$on their last birthdays was the same}$

That is, that $x$ and $y$ are the same age.

Then $\sim$ is an equivalence relation.

### Same Parents Relation

Let $P$ be the set of people.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { both of the parents of$x$and$y$are the same}$

Then $\sim$ is an equivalence relation.

### People with Same First Name

Let $P$ be the set of people.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text {$x$and$y$have the same first name}$

Then $\sim$ is an equivalence relation.

### Books with Same Number of Pages

Let $P$ be the set of books.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text {$x$and$y$have the same number of pages}$

Then $\sim$ is an equivalence relation.

### Even Sum Relation

Let $\Z$ denote the set of integers.

Let $\mathcal R$ denote the relation on $\Z$ defined as:

$\forall x, y \in \Z: x \mathrel {\mathcal R} y \iff x + y \text { is even}$

Then $\mathcal R$ is an equivalence relation.

The equivalence classes are:

$\eqclass 0 {\mathcal R}$
$\eqclass 1 {\mathcal R}$

## Also see

• Results about equivalence relations can be found here.