Section 3.3 Equivalence Relations 1

Section 3.3 Equivalence RelationSection 3.3 Equivalence RelationSection 3.3 Equivalence RelationSection 3.3 Equivalence Relationssss

Purpose of SectionPurpose of SectionPurpose of SectionPurpose of Section To introduce the concept of an equivalence relationequivalence relationequivalence relationequivalence relation and

show how it subdivides or partitions a set into distinct categories.

IntroductionIntroductionIntroductionIntroduction

Classifying objects and placing similar objects into groups provides a way

to organize information and focus attention on the similarities of like objects

and not on the dissimilarities of dislike objects. Mathematicians have been

classifying objects for millennia. Lines in the plane can be subdivided into

groups of parallel lines. Lines in a given group many not be the same line, but

they are considered equivalent insofar as our classification is concerned.

Triangles can also be classified as being congruent1 or noncongruent. In

modular arithmetic we say two integers are equivalent if they have the same

remainder when divided by a given number, say 7 for example. Here, the

numbers … -12, -5, 2, 9, 16, 23, 30, … would be considered equivalent,

whereas and 3 and 5 are not. This classification based on division by 7

subdivides the entire set of natural numbers into seven distinct equivalence

classes depending on whether a number’s remainder is 0,1,2,3,4,5, or 6 when

divided by 7. So what are the properties a relation must have in order that it

subdivide set into distinct categories? Keep reading.

Equivalence RelationEquivalence RelationEquivalence RelationEquivalence Relation

The reader is already familiar with one equivalence relation, the relation

of two things being “equal”. That’s right " "= . However, since ambiguity

restricts us from using the equal sign to denote the equivalence relation, we

have adopted the nearby symbol2 " "≡ . So what properties does an

equivalence relation possess?

Definition Definition Definition Definition A relation " "≡ on a set A is an equivalence relation equivalence relation equivalence relation equivalence relation if for all , ,x y z

in A , the following RST properties hold:

Reflexive:Reflexive:Reflexive:Reflexive: x x≡

Symmetric:Symmetric:Symmetric:Symmetric: if , then x y y x≡ ≡

Transitive:Transitive:Transitive:Transitive: if and , then x y y z x z≡ ≡ ≡ .

1 Recall that two triangles are congruent their corresponding sides and angles have the same measurements.

2 Another common notation for the equivalence relation is " "∼ .

Section 3.3 Equivalence Relations 2

Note:Note:Note:Note: Remember, a relation on a set A is a subset of all ordered pairs of A, so

keep in mind “ ≡” is a set. For example, ( ) means ,x y x y≡ ∈≡ . However, the

set notation looks awkward so we prefer to simply write x y≡ .

Example 1 Example 1 Example 1 Example 1 Equivalence Relations Equivalence Relations Equivalence Relations Equivalence Relations

The following relations are equivalence relations on the given sets.

▪ x y≡ means equality (“=”) between numbers or sets.

▪ x y≡ means “ x is congruent to y ” where ,x y are triangles.

▪ x y≡ means x y⇔ for logical sentences ,x y .

▪ x y≡ means “x has the same birthday as y” where x and y

are people.

▪ x y≡ means “ x differs from y by an integer multiple of 5”) on

the set � of integers.

Example 2 Non Equivalence RelationsExample 2 Non Equivalence RelationsExample 2 Non Equivalence RelationsExample 2 Non Equivalence Relations

The following relations are not equivalence relations on the given sets since

at least one of the three RST conditions fails.

▪ x y≡ means “x is in love with y” on the set of all people. (Not likely

symmetric; at least for two persons, one loves the other but not

Vice-versa.)

▪ x y≡ means " "x y≤ on the real numbers. (Not symmetric; 2 3≤ does

not imply 3 2≤ .)

▪ x y≡ means “ x and y have a common factor greater than 1” on the

set of integers. (Not transitive; 2 and 6 have a common factor, 6 and 3

have a common factor, but 2 and 3 do not.)

▪ x y≡ means “x ⊆ y” on a family of sets. (Not symmetric;

A B⊆ ⇒B A⊆ .)

Section 3.3 Equivalence Relations 3

Equivalence Broadens the Concept of Equality

Partitioning of Sets Partitioning of Sets Partitioning of Sets Partitioning of Sets into into into into Equivalence ClassesEquivalence ClassesEquivalence ClassesEquivalence Classes

The equivalence relation provides a way for mathematicians to subdivide

collections of objects into distinct subsets called partitions.

Definition: Definition: Definition: Definition: Let A be a set. A partitionpartitionpartitionpartition of A is a (finite or infinite) collection

{ }1 2, ,...A A of nonempty subsets of A such that

▪▪▪▪ the union of all thei

A is A

▪▪▪▪ the sets in the collection are pairwise disjoint; that is i j

A A∩ = ∅ for every

distinct pair and i j

A A .

Example Example Example Example 3333 The following are partitions of a set.

▪▪▪▪ The even and odd integers are a partition the integers into two distinct

sets.

Section 3.3 Equivalence Relations 4

▪ A collection of people partitioned into groups according to the first letter

in their surname.

▪ A set of students in a classroom partitioned into males and females. (If

there are no members of one sex, then the partition is the set itself.)

The following theorem lies at the heart of the importance of the equivalence

relation.

Theorem 1 Equivalence ClassesTheorem 1 Equivalence ClassesTheorem 1 Equivalence ClassesTheorem 1 Equivalence Classes

Let A be any set and " "≡ an equivalence relation on A. For every x A∈ ,

define a set, denoted by [ ]x , by

[ ] { }:x y A y x= ∈ ≡

This set is called the equivalequivalequivalequivalence classence classence classence class of x and consists of all elements of A

equivalent to x . Under these conditions each element of the set A belongs to

one and only one equivalence class, which means the equivalence relation

partitions the set A into distinct subsets.

Proof:Proof:Proof:Proof:

First observe every element x A∈ belongs to at least one equivalence

class since x x≡ , which means [ ]x x∈ . Also note the union of the equivalence

classes is A since every element x A∈ belongs to some equivalence class.

Now show the equivalence classes are disjoint. We do this by showing if

two equivalence classes intersect, then they are the same equivalence class.

Let ,s t A∈ and assume that the equivalence classes [ ]s and [ ]t which they

define, intersect. That is, there exists [ ] [ ]s ty ∈ ∩ (as drawn in the diagram).

Section 3.3 Equivalence Relations 5

We now show[ ] [ ]s t= . We first prove [ ] [ ]s t⊆ by letting [ ]sx ∈ and showing

that [ ]tx ∈ . Letting [ ]sx ∈ and using the symmetric and transitive properties

of the equivalence relation, we have

[ ]

[ ]

)

) but and so , hence

) since and we have

) but and so

) since and we have

i x s

ii y s y s s y

iii x s s y x y

iv y t y t

v x y y t x t

≡

∈ ≡ ≡

≡ ≡ ≡

∈ ≡

≡ ≡ ≡

.

This shows [ ]tx ∈ and hence [ ] [ ]s t⊆ . The proof [ ] [ ]s t⊇ is similar and so

[ ] [ ]s t= .

Significance of Significance of Significance of Significance of TheoremTheoremTheoremTheorem 1 (Partitioning Sets) 1 (Partitioning Sets) 1 (Partitioning Sets) 1 (Partitioning Sets)

An equivalence relation ≡ defined on a set A , partitions the set into

disjoint subsets, called equivalence classesequivalence classesequivalence classesequivalence classes. The elements of each equivalent

class are equivalent to each other, but not to any element in a different

equivalence class. One can think of the equivalence relation as a kind of

generalized equality where objects in a given class are “equal.” The set of

equivalences classes of A partitioned by ≡ is called the quotient setquotient setquotient setquotient set of A

modulo ≡ , and denoted by /A ≡ . See Figure 1.

Partitioning of A into Equivalence Classes

Figure 1

Example Example Example Example 4 (Modular Arithmetic)4 (Modular Arithmetic)4 (Modular Arithmetic)4 (Modular Arithmetic)

Let A = � be the integers. We define two numbers ,x y as equivalent if

they have the same remainder when divided by a given number, say 5, and we

Section 3.3 Equivalence Relations 6

write this as ( )mod 5x y≡ . Show this relation is an equivalence relation and

find the equivalence classes of this relation.

SolutionSolutionSolutionSolution

▪ reflexive:reflexive:reflexive:reflexive: ( )mod 5x x≡ since 5 divides 0x x− = .

▪ symmetric:symmetric:symmetric:symmetric: If ( )mod 5x y≡ then 5 divides x y− . This means there

exists an integer k such that 5x y k− = . But this implies

( )5 5y x k k− = − = − which means 5 divides y x− . Hence ( )mod 5y x≡ .

Hence ≡ is a symmetric relation.

▪ transitive:transitive:transitive:transitive: If ( )mod 5x y≡ and ( )mod 5y z≡ , then 5 divides

x y− and 5 divides y z− . Hence, there exist integers 1 2,k k that

satisfy 15x y k− = and 25y z k− = . Adding these equations yields

( ) ( ) 1 25 5x y y z k k− + − = +

or

( )1 2 35 5x z k k k− = + =

which shows that 5 divides x z− or ( )mod 5x z≡ . Hence ≡ is a

transitive relation.

The above equivalence relation divides the integers � into the equivalence

classes, where each equivalence class consists of integers with remainders of

0,1,2,3,4 when divided by 5. We give the numbers 0,1,2,3,4 special status and

assign them as “representatives of each equivalence class3 listed below.

TaTaTaTable 1 Equivalence Classes for ble 1 Equivalence Classes for ble 1 Equivalence Classes for ble 1 Equivalence Classes for ( )mod 5≡

[ ] { } { }

[ ] { } { }

[ ] { } { }

[ ] { } { }

[ ] { } { }

0 5 : 10, 5, 0, 5, 10

1 5 1: 9, 4, 1, 6, 11

2 5 2 : 8, 3, 2, 7, 12

3 5 3: 7, 2, 3, 8, 13

4 5 4 : 6, 1, 4, 9, 14

n n

n n

n n

n n

n n

= ∈ = − −

= + ∈ = − −

= + ∈ = − −

= + ∈ = − −

= + ∈ = − −

� � �

� � �

� � �

� � �

� � �

3 In number theory these equivalence classes are also called residue classes.

Section 3.3 Equivalence Relations 7

In this example, the quotient set / ≡� (set of equivalence classes) would be

[ ] [ ] [ ] [ ] [ ]{ }/ 0 , 1 , 2 , 3 , 4≡ =�

Note:Note:Note:Note: Some people do not understand how the remainder of 3 / 5− can be 2.

Remainders are defined as non negative integers, so we write

( )3 / 5 5 2 / 5 1 2 / 5− = − + = − + .

Modular ArithmeticModular ArithmeticModular ArithmeticModular Arithmetic:::: Modular arithmetic occurs in many areas of mathematics

as well as application in our daily lives. Instead of working with an infinite

number of integers, it is only necessary to work with a few. The smallest

number is ( )mod 2 , which subdivides the integers into two types, even and odd

integers. Public key cryptography allows computers to do exact arithmetic

( )mod n , where n is the product of two prime numbers each 100 digits long.

Example 5 (Equivalence Classes in the PlaneExample 5 (Equivalence Classes in the PlaneExample 5 (Equivalence Classes in the PlaneExample 5 (Equivalence Classes in the Plane))))

The Cartesian product A = � � defines points in the first quadrant of

the Cartesian plane with integer coordinates. Define a relationship between

( ),a b and ( ),c d as

( ) ( ), ,a b c d b a d c≡ ⇔ − = − .

Given that this relation defines an equivalence relation (See Problem 7), find

its equivalence classes.

SolutionSolutionSolutionSolution

This relation classifies points ( ),x y as equivalent if their y coordinate

minus their x coordinate are equal. This means two points (with positive

integer coefficients), are equivalent if they both lie on 45 degree lines

y x n= + , where n is an integer. Each value of n defines a different

equivalence class. Some typical equivalence classes are

Section 3.3 Equivalence Relations 8

Table Table Table Table 3333 Equivalence Classes Equivalence Classes Equivalence Classes Equivalence Classes

( ) { } ( )

( ) { } ( )

( ) { } ( )

( ) { } ( )

( ) { } ( )

... ...

2,0 (2,0), (3,1), (4, 2),... 2

1,0 (1,0), (2,1), (3,2),... 1

0,0 (0,0), (1,1), (2, 2),... 0

0,1 (0,1), (1, 2), (2,3),... 1

0, 2 (0,2), (1,3), (2, 4),... 2

n

n

n

n

n

= = −

= = −

= =

= =

= =

These equivalence classes are illustrated in Figure 2 as points with integer

coordinates lying on 45 degree lines in the first quadrant.

Equivalence Classes as Grid Points on Lines y x n= +

Figure 2

Margin NoMargin NoMargin NoMargin Note: te: te: te: “ a is equivalent to b ” means a is equal to b , not in every

respect, but as far as a particular property is concerned.

Graph of an Equivalence RGraph of an Equivalence RGraph of an Equivalence RGraph of an Equivalence Relation elation elation elation

The graph of the relation

( ) ( ) ( )( ) ( ) ( ) ( ) ( ){ }1,1 , 2, 2 , 3,3 4, 4 , 1,3 , 3,1 , 3,4 , 4,3R =

Section 3.3 Equivalence Relations 9

on the set { }1,2,3, 4A = is shown in Figure 3. It is an easy matter to verify that

R is an equivalence relation. What are the equivalence classes of this

relation?

SolutionSolutionSolutionSolution

The relation R says

1 1, 2 2, 3 3, 4 4, 1 3, 3 1, 3 4, 4 3≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡

Using the transitivity property, we can show 1 3 4≡ ≡ , and that the number 2 is

only equivalent to itself. Hence the set A has been subdivided into two

equivalence classes { }1,3, 4 and { }2 . Note that the two equivalence classes

are disjoint and their union is .A

Graph of an Equivalence Relation

Figure 3

Section 3.3 Equivalence Relations 10

ProblemsProblemsProblemsProblems

1. Let A be the set of students at a university and let x and y be students; i.e.

elements of A . Tell if the following relations on A are equivalent relations.

a) x is related to y iff x and y have the same major.

b) x is related to y iff x and y have the GPA.

c) x is related to y iff x and y are from the same country.

d) x is related to y iff x and y

e) x is related to y iff x and y have the same major.

2. (Equivalence Relations?)(Equivalence Relations?)(Equivalence Relations?)(Equivalence Relations?) Tell if the following relations R are equivalence

relations on a set A . If the relation is an equivalence relation, find the

equivalence classes.

a) xRy iff 2y x= ( )A = �

b) xRy iff x is a factor of y ( )A = �

c) xRy iff x and y are multiples of 5 ( )A = �

3. (Modular Arithmetic)(Modular Arithmetic)(Modular Arithmetic)(Modular Arithmetic) Given the set { }1,2,3,4,5,6,7,8A = and for ,x y A∈ we

have ( )mod 4x y≡ , i.e. 4 divides x y− . Prove that ≡ is an equivalence

relation and find the equivalence classes of the relation.

3. (True or False)(True or False)(True or False)(True or False)

a) ( )8 3 mod 5≡

b) ( )12 1 mod 6≡

c) ( )21 0 mod 3≡

d) ( )10 10 mod12≡

e) ( )99 9 mod 9≡

4. (Defining New Equivalence Relations)(Defining New Equivalence Relations)(Defining New Equivalence Relations)(Defining New Equivalence Relations) Let " " and " "≈∼ be two equivalence

relations on a set. Define two more relations by

( ) ( )

( ) ( )

a b a b a b

a b a b a b

⇔ ∨ ≈

⇔ ∧ ≈

� ∼

� ∼

Determine if " " and " "� � are equivalence relations.

Section 3.3 Equivalence Relations 11

5. (An Old Favorite) (An Old Favorite) (An Old Favorite) (An Old Favorite) The equals relation " "= is the most familiar equivalence

relation. What are the equivalence classes of the equals relation on the set

{ }1, 2,3,4,5A = ?

6. (Equivalence Relation). (Equivalence Relation). (Equivalence Relation). (Equivalence Relation) The Cartesian product A = � � defines the points

in the first quadrant with integer coordinates. Show the relationship between

( ),a b and ( ),c d defined by

( ) ( ), ,a b c d b a d c≡ ⇔ − = −

is an equivalence relation.

7. (Graph of an Equivalence Relation)(Graph of an Equivalence Relation)(Graph of an Equivalence Relation)(Graph of an Equivalence Relation) Figure 4 shows an equivalence relation

on the set { }1,2,3, 4A = . What are the equivalence classes?

Graph of an Equivalence Relation

Figure 4

8. (Projective Plane)(Projective Plane)(Projective Plane)(Projective Plane) Define ( ){ }2 0,0A = −� the set of all points in the plane

minus the origin. Define a relation between two points ( ),x y and ( ),x y′ ′ by

saying they are related if and only if they lie on the same line passing through

the origin. Show that this relation is an equivalence relation and draw a

graphical representation of the equivalence classes by picking a

representative from each class. The space of all equivalence classes under

this relation is called the projective planeprojective planeprojective planeprojective plane.

Section 3.3 Equivalence Relations 12

9. (Equivalence (Equivalence (Equivalence (Equivalence Classes Classes Classes Classes in Logic) in Logic) in Logic) in Logic) Define an equivalence relation on logical

sentences by saying two sentences are equivalent if they have the same truth

value. Find the equivalence classes in the following collection of sentences.

2

2 2

1 2 3

3 5

2 | 7

0 for some real number.

sin cos 1

Georg Cantor was born in 1845.

Leopold Kronecker was a big fan of Cantor.

Cantor's theorem guarantees larger and larger infinite sets.

x

x x

+ =

<

<

+ =

10. (Similar Matrices)(Similar Matrices)(Similar Matrices)(Similar Matrices) Two matrices ,A B if there is an invertible matrix

M such that 1MAM B− = .

a) Show that similarity of matrices is an equivalence relation.

b) Let

1 1 1 0

,4 1 1 1

A M

= =

find B . What common properties do ,A B share?

11. (Equivalence Relation on (Equivalence Relation on (Equivalence Relation on (Equivalence Relation on � )))) Let :f →� � . Define a relation on � by

( ) ( )a b f a f b≡ ⇔ =

a) Show ≡ is an equivalence relation.

b) What are the equivalence classes for ( ) 2f x x= ?

c) What are the equivalence classes for ( ) sinf x x= ?

12. (Arithmetic in Modular Arithmetic) Suppose

( )

( )

mod 5

mod 5

a c

b d

≡

≡

Show

Section 3.3 Equivalence Relations 13

( )

( )

( )

a) mod 5

b) mod 5

c) mod 5

a b c d

a b c d

ab cd

+ ≡ +

− ≡ −

≡

13. ((((Equivalent AnglesEquivalent AnglesEquivalent AnglesEquivalent Angles)))) We say two angles ,x y are related if and only if

2 2sin cos 1xRy x y⇔ + =

Show that R is an equivalence relation and identify the equivalence classes.

14. (Moduli in Modular Arithmetic)(Moduli in Modular Arithmetic)(Moduli in Modular Arithmetic)(Moduli in Modular Arithmetic) Note that ( )2 2 0 mod 4⋅ = but neither factor

is zero. What is 2 4⋅ modulo 8? Is there any product a b⋅ with neither factor

0, but ( )0 mod3a b⋅ ≡ ? Can you form a conjecture about the moduli related to

this phenomenon?