# Equivalence Relation/Examples

## Examples of Equivalence Relations

### 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:

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

### Months that Start on the Same Day of the Week

Let $M$ be the set of months of the year according to the (usual) Gregorian calendar.

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

$\forall x, y \in M: x \sim y \iff \text {$x$and$y$both start on the same day of the week}$

Then $\sim$ is an equivalence relation.

## Examples of Non-Equivalence Relations

### Different 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$on their last birthdays was not the same}$

Then $\sim$ is not an equivalence relation.

### Is the Mother Of is not Equivalence

Let $P$ denote 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$is the mother of$y$}$

Then $\sim$ is not an equivalence relation.

### Is the Sister Of is not Equivalence

Let $P$ denote 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$is the sister of$y$}$

Then $\sim$ is not an equivalence relation.

### Common Ancestor Relation

Let $P$ be the set of people.

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

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text {$x$and$y$have an ancestor in common}$

Then $\sim$ is not an equivalence relation.

### Greater Than is not Equivalence

Let $\R$ denote the set of real number.

Let $>$ denote the usual relation on $\R$ defined as:

$\forall \tuple {x, y} \in \R \times \R: x > y \iff \text {$x$is (strictly) greater than$y$}$

Then $>$ is not an equivalence relation.

### $\forall x, y \in \R: x + y \in \Z$ is not Equivalence

Let $\R$ denote the set of real numbers.

Let $\sim$ denote the relation defined on $\R$ as:

$\forall \tuple {x, y} \in \R \times \R: x \sim y \iff x + y \in \Z$

Then $\sim$ is not an equivalence relation.

### Divisor Relation is not Equivalence

Let $\Z$ denote the set of (strictly) positive integers.

Let $x \divides y$ denote that $x$ is a divisor of $y$

Then $\divides$ is not an equivalence relation.

### Sum of Integers is Divisible by $3$ is not Equivalence

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 divisible by$3$}$

Then $\mathcal R$ is not an equivalence relation.