Section 3.1 Relations 1

"When two objects, qualities, classes, or attributes, viewed together by the mind, are

seen under some connection, that connection is called a relation."

… Augustus De Morgan

Section 3.1 Relations Section 3.1 Relations Section 3.1 Relations Section 3.1 Relations

Introduction Introduction Introduction Introduction

In mathematics the word relation is used to show the existence or

nonexistence of certain connections pairs of objects, taken in definite order,

as in

“is less than” “is a subset of”

“is perpendicular to” “divides”

“is greater than” “is congruent to”

“is parallel to” “is equivalent to”

There are several types of relations, such as order relations, equivalence

relations, functions, and graph/networks, all of which play important roles in

various areas of mathematics, and are studied in this chapter.

Cartesian Product of Sets Cartesian Product of Sets Cartesian Product of Sets Cartesian Product of Sets

Before formally defining a relation, we must introduce the concept of the

Cartesian product of two sets.

Example 1Example 1Example 1Example 1 Let { }1, 2A = and { }1,3,4B = , then

Purpose of SectionPurpose of SectionPurpose of SectionPurpose of Section To introduce the concept of a relationrelationrelationrelation between objects,

the objects being almost anything one can imagine. This section acts as the

backdrop for the three important relations studied in the following sections;

the order relation, the equivalence relation, function relation, and graphs.

Definition Definition Definition Definition Cartesian P Cartesian P Cartesian P Cartesian Product:roduct:roduct:roduct:

Let A and B be arbitrary sets. The Cartesian Cartesian Cartesian Cartesian pppproductroductroductroduct of A and B,

denoted A B× and read ” cross A B ” is the set of ordered pairs

( ){ }, : ,A B a b a A b B× = ∈ ∈ .

That is A B× is the set of all ordered pairs ( ),a b , where the first coordinate

a is taken from A and the second coordinate b is taken from B.

Section 3.1 Relations 2

( ) ( ) ( ) ( ) ( ) ( ){ }1,1 , 1,3 , 1, 4 , 2,1 , 2,3 , 2,4A B× =

If we represent the points ( ),a b as points in the xy -Cartesian plane, the

Cartesian product A B× consists of the points displayed in Figure 1.

Cartesian Product A B×

Figure 1

Order is important for Cartesian products. For the same sets ,A B , we have

( ) ( ) ( ) ( ) ( ) ( ){ }1,1 , 1, 2 , 3,1 , 3,2 , 4,1 , 4,2B A× =

which illustrates that in general A B B A× ≠ × . The points in the Cartesian

product B A× are reflections of the points in A B× through the line y x= .

Cartesian Product B A×

Figure 2

MarMarMarMargin Note: gin Note: gin Note: gin Note: Do not confuse the ordered pair ( )0,1 from the open interval on

the real line ( ) { }0,1 : 0 1x x= < < .

Section 3.1 Relations 3

Cartesian products on more than two sets are defined accordingly. For

instance, for three sets A, B, and C we would define

( ){ }, , : , ,A B C a b c a A b B c C× × = ∈ ∈ ∈ .

When the coordinates are real numbers, the ordered triple ( ), ,a b c can

interpreted as points in three-dimensional space.

The Cartesian product interacts with the union and intersection of sets

by the following properties.

Example Example Example Example 2222 If A, B and C are sets, then

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

)

)

)

)

a A B C A B A C

b A B C A B A C

c A B C D A C B D

d A B C D A C B D

× ∪ = × ∪ ×

× ∩ = × ∩ ×

× ∩ × = ∩ × ∩

× ∪ × = ∪ × ∪

Proof:Proof:Proof:Proof:

We prove a). The proofs of b), c) and d) are left to the reader. To prove

part a) note that both sets ( )A B C× ∪ and ( ) ( )A B A C× ∪ × are sets of

ordered pairs, where their elements are of the form ( ),x y . Normally, a proof

showing the equality of two sets requires two separate proofs, where one

proves each set is a subset of the other, but in this case we can go “forward”

and “backwards” at the same time. Watch carefully.

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( ) ( )

, and

and or

and or and

, or ,

, .

x y A B C x A y B C

x A y B y C

x A y B x A y C

x y A B x y A C

x y A B A C

∈ × ∪ ⇔ ∈ ∈ ∪

⇔ ∈ ∈ ∈

⇔ ∈ ∈ ∈ ∈

⇔ ∈ × ∈ ×

⇔ ∈ × ∪ ×

Margin Note: Margin Note: Margin Note: Margin Note: Cartesian products are also called ddddirect productsirect productsirect productsirect products or ccccross ross ross ross

productsproductsproductsproducts.

Section 3.1 Relations 4

Visualization of ( ) ( ) ( )A B C A B A C× ∪ = × ∪ ×

Figure 2

Since all the logical connectives in the proof are “if and only if” you can start

either at the top or bottom of the argument and work to the other end. Many

proofs showing the equality of sets work in this manner. Figure 2 provides a

visualization of this identity although in general , ,A B C need not be subsets of

the real numbers.

MargMargMargMargin Note:in Note:in Note:in Note: Readers are already familiar with the most popular Cartesian

product, which is 2≡ ×� � � , the Cartesian plane.

In Figure 3 we have drawn some typical Cartesian products.

Section 3.1 Relations 5

Typical Cartesian Products

Figure 3

RelationsRelationsRelationsRelations

A (binary) relationrelationrelationrelation is a function that assigns truth values (true or false) to

two individuals, which can be numbers, sets, functions and so on. When the

value assigned is true, we say the individuals are related, like if the relation

were “is less then” then the pair of numbers ( )2,3 would be assigned “true”

since 2 3< . The reader is already familiar with many relations, such as the

equal relation " "= which would assign a truth value to the pair ( )3,3 but not

the pair ( )3, 4 . Relations also occur outside of mathematics as the following

relations.

John is taller than Jerry

Susan is the daughter of John

Harry loves Susan

Des Moines is a city in Iowa

i

i

i

i

Section 3.1 Relations 6

Relations can be between numbers and themselves, like 2 3< , between people

and themselves, and between a city and a state.

John Jerry "is taller than"

2 3 " "

Susan John "is the daughter of "

R R

R R

R R

=

= <

=

i

i

i

This motivates the general definition of a relation.

DefinitionDefinitionDefinitionDefinition:::: RelationRelationRelationRelation

Let A and B be sets and A B× the Cartesian product. A binarybinarybinarybinary relationrelationrelationrelation

from the set A to the set B , which we denote by R , is a subset of A B× . If

( , )x y R∈ we denote this by writing xRy , where we say “ x and y are related

( relative to )R . If ( ),x y R∉ we say that x and y are not related, this denote

this by xRy . In case A B= a relation from A to A is called a relation on A.

The elements in a relation R is called the graphgraphgraphgraph of the relation.

You can think of a binary relation as an operation that assigns either T or

F (or 1 or 0) to the pair of objects. If the value is true we say the objects are

related, if the value is false the objects are not related.

Margin Note:Margin Note:Margin Note:Margin Note: The definition of a relation can be a bit confusing at first,

thinking of a relation as both a set (i.e. subset of a Cartesian product) and as

a relation between things, like “=”, “<”, “loves” and so on.. Just realize that if

a point ( ),x y R∈ then we often denote this by writing xRy .

Margin Note:Margin Note:Margin Note:Margin Note: ( ) if and only if ,xRy x y R∈

The following examples will familiarize you with relations.

Example Example Example Example 3 (Typical Relation)3 (Typical Relation)3 (Typical Relation)3 (Typical Relation)

Consider a set A of five students and a set B of four movies:

{ }

{ }

Mary, John, Ann, Sally, Jim

Crash, Batman, Star Wars III, Sin City

A

B

=

=

and consider the relation R A B⊆ × defined by

Section 3.1 Relations 7

likes movie yxRy x=

and consists of the 10 x ’s drawn in the 20 boxes in Table 1. The elements of

R describe those students who like a certain movie. Since

( )John, Sin City R∈ , this means John likes the movie Sin City. Since

( )Ann, Star Wars IX R∉ , this means Ann does not like Star Wars IX. The

subset R A B⊆ × is a relation on A B× (read “relation from A to B ”).

Sin CitySin CitySin CitySin City x x

Star Wars IXStar Wars IXStar Wars IXStar Wars IX x x x

BatmanBatmanBatmanBatman x x

CrashCrashCrashCrash x x x

MaryMaryMaryMary JohnJohnJohnJohn AnnAnnAnnAnn SallySallySallySally JimJimJimJim

Relation “likes movie” on { } { }Students Movies×

Table 1

Example Example Example Example 4444 Identify the Relation Identify the Relation Identify the Relation Identify the Relation

The Cartesian product of { } { }1,2 , 1,3,4A B= = consists of the 2 3 6⋅ =

pairs of points

( ) ( ) ( ) ( ) ( ) ( ){ }1,1 , 1, 2 , 1,3 , 2,1 , 2, 2 , 2,3A B× =

A relation on A B× is any one of the 62 64= subsets of A B× . A popular

relation (do you recognize it?) from A to B would be

( ) ( ) ( ){ }1,2 , 1,3 , 2,3R A B= ⊆ ×

In other words, “1 is related to 2”, 1 is related to 3” and “2 is related to 3”.

Do you recognize the relation now? We would also denote this relation by

writing

( )

( )

( )

1 2 since 1,2 more commonly (1 2)

1 3 since 1,3 more commonly (1 3)

2 3 since 2,3 more commonly (2 3)

R R

R R

R R

∈ <

∈ <

∈ <

In this example the relation R is the relation “is strictly less than” which we

normally denote by " "< . We also write1 R1 or 1< 1 (1 is not related to itself)

since 1 is not strictly less than 1. The relation is illustrated by a “directed

graph” drawn in Figure 4.

Section 3.1 Relations 8

Graphical Representations of the Relation " "< on A B× .

Figure 4

Example 5 Subset and Membership RelationsExample 5 Subset and Membership RelationsExample 5 Subset and Membership RelationsExample 5 Subset and Membership Relations

The subset " "⊆ is a relation on the power set ( )P A of a set A , and

" "∈ is a relation from the set A to its power set ( )P A .

Example Example Example Example 6666 A A A A More More More More Familiar RelationFamiliar RelationFamiliar RelationFamiliar Relation

Often A and B are subsets of the real numbers � such as

{ } [ ]: 0 1 0,1A B x x= = ∈ ≤ ≤ =�

Here the Cartesian product is the unit square

[ ] [ ] ( ){ }0,1 0,1 ( , : 0 1,0 1x y x y× = ≤ ≤ ≤ ≤

which is shaded in Figure 5. A common relation from [ ]0,1 to itself is

[ ] [ ]{ }2( , ) 0,1 0,1 :R x y y x= ∈ × =

which the reader will recognize as the graph of the parabola 2y x= drawn in

Figure 5. We would call this relation the “squaring” relation and since

( )1/ 2,1/ 4 R∈ we would say 1/ 2 is related to 1/ 4 , the denote this relationship

Section 3.1 Relations 9

by 2(1/ 2) 1/ 4= . Later, we will denote function relations by , ,...f g and so

this relation would be denoted by

( ) [ ] [ ]{ }2, 0,1 0,1 :f x y y x= ∈ × = .

A Function Relation

Figure 5

Margin Note:Margin Note:Margin Note:Margin Note: A relation R from the real numbers � to itself is a subset of

the Cartesian plane ×� � . For any point ( ),x y R∈ we say that the first

coordinate x is related to the second coordinate y .

Inconsistency in Notation: Inconsistency in Notation: Inconsistency in Notation: Inconsistency in Notation: Although relations are sets, we often do not

“think” of them that way. For instance, we express the inequality relation ≤

by writing xRy as in 2 3≤ . The equivalent set notation for ≤ would be the

awkward looking ( )2,3 ∈ ≤ . (You would probably scratch you head if

inequalities were written like that!) Also, we seldom use the xRy “form” to

express function relations. They have their own notion. For example, we

normally write functions like 2( )y f x x= = . This relation written as xRy

would be 2xfx , again an awkward looking expression for most of us.

Sometimes, however, it is convenient to express functions in set notation, as

when we write ( )2,x x f∈ . This notation has merit since it interprets a

function in terms of what we normally call the graph of the function.

Higher Order Higher Order Higher Order Higher Order RelationsRelationsRelationsRelations:::: In set theory and logic a nnnn----ary ary ary ary is a relation that assigns

a truth value (true or false) to k members of a given set. A ternaryternaryternaryternary relation is

a relation ( ), ,R a b c that assigns a truth value to three members. For example

the statement that for nonzero integers , ,a b c

Section 3.1 Relations 10

3 3 3a b c+ =

is a ternary relation on � , which according to Andrew Wiles1 is false.

.

Domain and Range of a RelationDomain and Range of a RelationDomain and Range of a RelationDomain and Range of a Relation

Two important aspects of relations are the domain and range of a

relation.

DefinitionDefinitionDefinitionDefinition: Domain and Range: Domain and Range: Domain and Range: Domain and Range Let R be a relation from A to B . The domaindomaindomaindomain

of the relation is the set

( ) { }Dom : such that .R x A y B xRy= ∈ ∃ ∈

The rangerangerangerange of the relation R is

( ) { }Range : such that .R y B x A xRy= ∈ ∃ ∈

In Plain English: The domain of a relation R from to A B is the set of first

coordinates of the ordered pairs in R . The range of a relation R is the set of

second coordinates of the ordered pairs of R . By definition, we have

( ) ( )Dom , RangeR A R B⊆ ⊆ .

Example Example Example Example 7777 Let A B= = � (real numbers). We define a relation on � � by

( ){ }2 2, : 1R x y x y= ∈ × + ≤� � which are points on and inside the unit circle in

the Cartesian plane. Here

( ) { }

( ) { }

Dom : 1 1

Range : 1 1

R x x

R y y

= ∈ − ≤ ≤

= ∈ − ≤ ≤

�

�

.

Margin Note:Margin Note:Margin Note:Margin Note: If ( ),a b R∈ , then

i) ( )Doma R∈

ii) ( )Rangeb R∈

iii) is related to a b

See Figure 6.

1 English mathematician Andew Wildes verified Format’s Last Theorem in the affirmative by proving that

among nonzero integers , ,a b c the equation n n n

a b c+ = has no solutions for integers 3n ≥ .

Section 3.1 Relations 11

Relation ( ){ }2 2, : 1R x y x y= + ≤

Figure 6

Inverses and Compositions Inverses and Compositions Inverses and Compositions Inverses and Compositions

Two common ways to construct new relations from old ones are inverse inverse inverse inverse

relationsrelationsrelationsrelations and compositioncompositioncompositioncomposition of relations.

DefinitionDefinitionDefinitionDefinition If R is a relation from A to B , then the inverseinverseinverseinverse of R is defined as

( ) ( ){ }1 , : ,R y x x y R−

= ∈ .

For example the inverse of ( ) ( ) ( ){ }1,2 , 3,5 , 4,1R = is ( ) ( ) ( ){ }1 2,1 , 5,3 , 1, 4R−

= .

Example Example Example Example 8888

Let { }1,2,3, 4A = with a relation on A given by

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

The inverse of R is ( ) ( ) ( ) ( ){ }1 2,1 , 3,1 , 4, 2 , 4,3R−

= . Both R and 1R− are drawn

in Figure 7. Note that the graph of 1R− is the mirror image or reflection of R

across the line y x= .

Section 3.1 Relations 12

Graphs of R and 1R−

Figure 7

Example Example Example Example 9999 (Inverse of a Curve) (Inverse of a Curve) (Inverse of a Curve) (Inverse of a Curve)

Given the relation on � defined by ( ){ }2, :R x y y x= = the inverse is

( ){ }1 2, :R x y x y−= = which we have drawn in Figure 8.

Graphs of R and 1R−

Figure 8

Composition of RelationsComposition of RelationsComposition of RelationsComposition of Relations

The composition of two (or more) relations is similar to the composition

of functions of which the reader is familiar. Recall that the composition of

functions ( )f x and ( )g x , written f g� , is defined by ( )( ) ( )( )f g x f g x=� for

all x in the domain of g and ( )g x in the domain of f . The generalization of

this definition to relations is as follows.

Section 3.1 Relations 13

Definition:Definition:Definition:Definition: Let R be a relation from A to B (i.e. a subset of A B× ) and S a

relation from B to C (i.e. a subset of B C× ). The compositecompositecompositecomposite (or compositioncompositioncompositioncomposition)

of R and S is the relation

( ) ( ) ( ){ }, : there exists a such that , and ,S R a c b B a b R b c S= ∈ ∈ ∈� .

Example Example Example Example 10101010 (Composition of Relations) (Composition of Relations) (Composition of Relations) (Composition of Relations)

Let { } { }1,2,3, 4 , , , , {cat, dog, horse}A B a b c C= = = and define the relations

( ) ( ) ( ) ( ) ( ){ }

( ) ( ) ( ) ( ) ( ){ }

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

,dog , ,horse , ,cat , ,dog , ,horse

R a c a b b

S a b b c c

=

=

which are illustrated in Figure 9. The composition S R� in this diagram

consists of all “paths” starting from A and ending at C , which are

( ) ( ) ( ) ( )( ) ( ){ }1,dog , 2,dog , 3,cat , 3,horse 4,cat , 4,horseS R =�

Composition S R�

Figure 9

Section 3.1 Relations 14

ProblemsProblemsProblemsProblems

1. 1. 1. 1. (Four Basic Cartesian Products)(Four Basic Cartesian Products)(Four Basic Cartesian Products)(Four Basic Cartesian Products) Let { } { }1,2,3 , ,A B a b= = . Find the

following Cartesian products.

a) A B×

b) B A×

c) A A×

d) B B×

2. . . . (Cartesian Products)(Cartesian Products)(Cartesian Products)(Cartesian Products) For each pair of sets A and B find the Cartesian

products and A B B A× × .

a) { } { }0, 2 , 1,0,1A B= = −

b) { } { }, , , ,A a b B a b c= =

c) ,A B= =� �

d) ,A B= =� �

e) { }, 1,0,1A B= = −�

f) ,A B= =� �

3. (Graphing a Relation)(Graphing a Relation)(Graphing a Relation)(Graphing a Relation) Sketch the following relations.

a) ( ){ }2 2, : 1R x y x y= ∈ × + =� �

b) ( ){ }, : sinR x y y x= ∈ × =� �

c) ( ){ }2, :R x y x y= ∈ × =� �

d) ( ){ }, : 1, 1R x y x y= ∈ × ≤ ≥� �

e) ( ) ( ) ( ){ }, : 0 mod3 , 1 mod3R x y x y= ∈ × ≡ ≡� �

f) ( ){ }, : divides R x y x y= ∈ ×� �

4. (Algebra of Relations)(Algebra of Relations)(Algebra of Relations)(Algebra of Relations) Suppose [ ] [ ] [ ]0,3 , 2,5 , 1, 4A B C= = = are closed

intervals on the real � . Sketch the following relations on � � .

a) ( )A B C∪ ×

b) ( )A B C∩ ×

c) ( ) ( )A B A C× ∪ ×

d) ( )A A C× ∩

e) ( )A A B× ∩

Section 3.1 Relations 15

5. (Naming a Relation)(Naming a Relation)(Naming a Relation)(Naming a Relation) Find the inverse relations of the following relations on

{ }1,2,3A = . Plot the points of the relation and give each relation a name which

best describes it.

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

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

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

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

6. (Inverse Relations on (Inverse Relations on (Inverse Relations on (Inverse Relations on � )))) Find and graph the inverse relations of the

following relations on � .

a) ( ){ }, : logaR x y y x= =

b) ( ){ }, : 2 0R x y x y= + =

c) ( ){ }, : 1R x y xy= =

d) ( ){ }, : 1R x y x y= + ≤

e) ( ){ }, : 1R x y x y= − ≤

f) ( ){ }, : 1R x y x y= + ≤

7. (Cartesian Product Identities)(Cartesian Product Identities)(Cartesian Product Identities)(Cartesian Product Identities) Prove the following.

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

)

)

)

a A B C A B A C

b A B C D A C B D

c A B C D A C B D

× ∩ = × ∩ ×

× ∩ × = ∩ × ∩

× ∪ × = ∪ × ∪

8. (N(N(N(Number of Relationumber of Relationumber of Relationumber of Relations)s)s)s) If a set A has m elements and B has n elements

then show there are 2mn distinct relations from A to B.

9. (I(I(I(Identity Relatiodentity Relatiodentity Relatiodentity Relation)n)n)n) For any set A, the set { }( , ) :AI x x x A= ∈ is called the

identity relation on A. For the set { }1,2,3, 4A = , what is the domain and range

of A

I and draw its graph.

10 (Properties of Relations)(Properties of Relations)(Properties of Relations)(Properties of Relations) Determine whether the following relations are

reflexive, anti symmetric, or transitive.

a) ( ), the greatest common divisor of and is 1x y R x y∈ ⇔

Section 3.1 Relations 16

b) ( ),x y R x y∈ ⇔ <

c) ( ), 5 divides x y R x y∈ ⇔ −

d) ( ),x y R x y∈ ⇔ ≥

11. Give examples of relations on the { }1,2,2,4,5A = having the following

properties.

a) Reflexive, antisymmetric, not transitive

b) Reflexive, symmetric, not transitive

c) Reflexive, antisymmetric, transitive

d) Not reflexive, symmetric, transitive