﻿ Equivalence Relations

# Equivalence Relations

Suppose students in a class are getting together in groups to do an activity. The teacher wants to make sure of two things:

• no student is in more than one group (i.e., the groups don't overlap), and
• every student is accounted for (i.e., all the groups, together, make up the entire class).

From a mathematical viewpoint, what has been created is a partition of the class:

DEFINITION partition
A partition of a set $\,S\,$ is a collection of non-overlapping subsets of $\,S\,$ that, together, make up all of $\,S\,.$

More precisely, ‘make up all of $\,S\,$’ means that the union of the subsets is $\,S\,.$

A partition of a set $\,S\,$

## Examples

Let $\,S=\{1,2,3,4,5\}\,.$

The sets $\,\{1,2,3\}\,$ and $\,\{4,5\}\,$ form a partition of $\,S\,$: they don't overlap, and, together, they make up all of $\,S\,.$

The sets $\,\{1,5\}\,$ and $\,\{2,3,4\}\,$ form a different partition of $\,S\,.$

The sets $\,\{1\}\,$ and $\,\{3\}\,$ and $\,\{2,4,5\}\,$ form a different partition of $\,S\,.$

The sets $\,\{1,2\}\,$ and $\,\{3,4\}\,$ do not form a partition of $\,S\,$; together, they don't make up all of $\,S\,.$

The sets $\,\{1,2,3\}\,$ and $\,\{3,4,5\}\,$ do not form a partition of $\,S\,$; they overlap—the number $\,3\,$ is in more than one of the subsets.

The sets $\,\{1,2\}\,$ and $\,\{2,3\}\,$ and $\,\{4,5\}\,$ do not form a partition of $\,S\,$; they overlap—the number $\,2\,$ is in more than one of the subsets.

## Being ‘Related To Each Other’ In a Set

One of the most important ways in math to create a partition of a set is by using an equivalence relation, which is the subject of this section.

Given a set $\,S\,,$ we first need a way to talk about members of the set being related to each other. The symbol ‘$\,\sim\,$’ is used for ‘is related to’. The sentence ‘$\,x\sim y\,$’ is read as ‘$\,x\,$ is related to $\,y\,$’.

The concept is best illustrated with some examples:

### Same-Sex Relation

Let $\,S\,$ be the set of all people, and let $\,x\,$ and $\,y\,$ be members of $\,S\,.$ Define:

$\,x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same sex

Suppose that Carol and Julia are female; Rick and Karl are male.

 TRUE SENTENCE READ AS: WHY TRUE? Carol $\,\sim \,$ Julia Carol is related to Julia both Carol and Julia are female Carol $\,\not\sim\,$ Karl Carol is not related to Karl Carol and Karl do not have the same sex Rick $\,\sim \,$ Karl Rick is related to Karl both Rick and Karl are male

### Same-Remainder Relation

Let $\,S=\{0,1,2,3,\ldots\}\,.$

For $\,x\,$ and $\,y\,$ in $\,S\,,$ define:

$x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same remainder upon division by $\,3\,$

 TRUE SENTENCE READ AS: WHY TRUE? $\,5\sim 8\,$ $\,5\,$ is related to $\,8\,$ When $\,5\,$ is divided by $\,3\,,$ the remainder is $\,2\,$: $$\cssId{s69}{5 = 1\cdot 3 \color{red}{+ 2}}$$ When $\,8\,$ is divided by $\,3\,,$ the remainder is $\,2\,$: $$\cssId{s71}{8 = 2\cdot 3 \color{red}{+ 2}}$$ Thus, both have the same remainder ($\,2\,$) when divided by $\,3\,.$ $\,3\sim 12\,$ $\,3\,$ is related to $\,12\,$ When $\,3\,$ is divided by $\,3\,,$ the remainder is $\,0\,$: $$\cssId{s76}{3 = 1\cdot 3 + 0}$$ When $\,12\,$ is divided by $\,3\,,$ the remainder is $\,0\,$: $$\cssId{s78}{12 = 4\cdot 3 + 0}$$ Thus, both have the same remainder ($\,0\,$) when divided by $\,3\,.$ $\,1\not\sim 11\,$ $\,1\,$ is not related to $\,11\,$ When $\,1\,$ is divided by $\,3\,,$ the remainder is $\,1\,$: $$\cssId{s83}{1 = 0\cdot 3 + 1}$$ When $\,11\,$ is divided by $\,3\,,$ the remainder is $\,2\,$: $$\cssId{s85}{11 = 3\cdot 3 + 2}$$ Thus, $\,1\,$ and $\,11\,$ have different remainders when divided by $\,3\,.$

### Equivalent-Fraction Relation

Let $\,S\,$ be the set of all ordered pairs of positive integers.

For $\,(a,b)\,$ and $\,(c,d)\,$ in $\,S\,,$ define:

$\,(a,b) \sim (c,d) \,$
if and only if
$\,ad=bc\,$

 TRUE SENTENCE READ AS: WHY TRUE? $\,(1,3)\sim (2,6)\,$ $\,(1,3)\,$ is related to $\,(2,6)\,$ $\,1\cdot 6 = 3\cdot 2\,$ $\,(2,5)\sim (4,10)\,$ $\,(2,5)\,$ is related to $\,(4,10)\,$ $\,2\cdot 10 = 5\cdot 4\,$ $\,(1,3)\not\sim (2,5)\,$ $\,(1,3)\,$ is not related to $\,(2,5)\,$ $\,1\cdot 5\neq 3\cdot 2\,$

(This concept is very familiar to you; it's just being cast into an unfamiliar setting. Hint—think about fractions!)

## Equivalence Relations

Now, we are in a position to define an equivalence relation:

DEFINITION equivalence relation
Let $\,S\,$ be a set.

Then, $\,\sim \,$ is an equivalence relation on $\,S\,$ if and only if the following properties are satisfied for all members $\,x\,,$ $\,y\,$ and $\,z\,$ in $\,S\,$:

• REFLEXIVE property:  $\,x\sim x\,$

(every member is related to itself)

• SYMMETRY property:  if $\,x\sim y\,,$ then $\,y\sim x\,$

(if one member is related to another, then the other is related to the one)

• TRANSITIVE property:  if $\,x\sim y\,$ and $\,y\sim z\,,$ then $\,x\sim z\,$

(if one is related to another, and this other is related to a third, then the first is related to the third)

## Example (An Equivalence Relation on the Set of All People)

(1)  Let $\,S\,$ be the set of all people, and let $\,x\,$ and $\,y\,$ be members of $\,S\,.$ Define:

$\,x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same sex

Then, $\,\sim \,$ is an equivalence relation on $\,S\,,$ as follows:

• Reflexivity: every person has the same sex as his/her self
• Symmetry: if a person has the same sex as another person, then the other person has the same sex as the first
• Transitivity: if person #1 has the same sex as person #2, and person #2 has the same sex as person #3, then person #1 has the same sex as person #3

## Example (An Equivalence Relation on the Set of All Nonnegative Integers)

(2)  Let $\,S=\{0,1,2,3,\ldots\}\,.$

For $\,x\,$ and $\,y\,$ in $\,S\,,$ define:

$\,x\sim y\,$
if and only if
$\,x\,$ and $\,y\,$ have the same remainder upon division by $\,3\,$

Then, $\,\sim \,$ is an equivalence relation on $\,S\,,$ as follows:

• Reflexivity: every number has the same remainder upon division by $\,3\,$ as itself
• Symmetry: if a number has the same remainder upon division by $\,3\,$ as another, then the other has the same remainder upon division by $\,3\,$ as the first
• Transitivity: if a number has the same remainder upon division by $\,3\,$ as another, and this other has the same remainder upon division by $\,3\,$ as a third, then the first number has the same remainder upon division by $\,3\,$ as the third

## Example(NOT An Equivalence Relation)

(3)  Let $\,S=\{0,1,2,3,\ldots\}\,.$

For $\,x\,$ and $\,y\,$ in $\,S\,,$ define:

$\,x\sim y\,$
if and only if
$\,x\lt y\,$

Then, $\,\sim \,$ is not an equivalence relation on $\,S\,.$ It fails both reflexivity (since $\,x\,$ is not less than $\,x\,$) and symmetry (if $\,x\,$ is less than $\,y\,,$ then $\,y\,$ is not less than $\,x\,$).

## Equivalence Classes

Once you select a member of a set which has an equivalence relation on it, you often want to study all the members which are related to it. This leads us to:

DEFINITION equivalence class

Let $\,\sim \,$ be an equivalence relation on a set $\,S\,.$ Let $\,x\,$ be a member of $\,S\,.$

The equivalence class determined by $\,x\,$ is the set of all members of $\,S\,$ that are related to $\,x\,.$

Here is one key reason why equivalence relations are so important:

The equivalence classes of a set always form a PARTITION of the set.

For Example (1) above, there are only two equivalence classes:   males and females.

For Example (2) above, there are three equivalence classes:

• $\,\{0,3,6,\ldots\}\,$

(numbers that are divisible by $\,3\,$; that is, the remainder is $\,0\,$)

• $\,\{1,4,7,\ldots\}\,$

(numbers that leave a remainder of $\,1\,$ upon division by $\,3\,$)

• $\,\{2,5,8,\ldots\}\,$

(numbers that leave a remainder of $\,2\,$ upon division by $\,3\,$)

As a final example, congruence is an equivalence relation on the set of all geometric figures. Let $\,G_1\,,$ $\,G_2\,,$ and $\,G_3\,$ be geometric figures. Then:

• Reflexivity: $\,G_1 \cong G_1$ (every geometric figure is congruent to itself)
• Symmetry: if $\,G_1\cong G_2\,$ then $\,G_2\cong G_1$
• Transitivity: if $\,G_1\cong G_2\,$ and $\,G_2\cong G_3\,,$ then $\,G_1\cong G_3$