# Equivalence Relation/Examples

## Examples of Equivalence Relations

### Equality

Let $S$ be a set.

Let the relation $\RR$ on $S$ be defined as:

- $\forall x, y \in S: x \mathrel \RR y \iff x = y$

that is, the equality relation on $S$.

Then $\RR$ is an equivalence relation.

### 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 $\RR$ denote the relation on $\Z$ defined as:

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

Then $\RR$ is an equivalence relation.

The equivalence classes are:

- $\eqclass 0 \RR$
- $\eqclass 1 \RR$

### 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.

### Points on Same Horizontal Line

Equivalence Relation/Examples/Points on Same Horizontal Line

### $z^4 = w^4$ on Complex Numbers

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the relation on $\C$ defined as:

- $\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is an equivalence relation.

### $\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ on Integers

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

- $\forall x, y \in \Z: x \mathrel \RR y \iff \sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$

Then $\RR$ 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 $\Z_{>0}$ 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 $\RR$ denote the relation on $\Z$ defined as:

- $\forall x, y \in \Z: x \mathrel \RR y \iff x + y \text { is divisible by } 3$

Then $\RR$ is not an equivalence relation.