MTH-4109-1
S
ets, Relations
and
Functions
SETS,
RELATIONS
AND
FUNCTIONS
MTH-4109-1
Mathematics Project Coordinator: Jean-Paul Groleau
Authors: Serge Dugas, Louise Allard
Content Revision: Jean-Paul Groleau, Alain Malouin
Pedagogical Revision: Jean-Paul Groleau
Translation: Claudia de Fulviis
Linguistic Revision: Johanne St-Martin
Electronic Publishing: P.P.I. Inc.
Cover Page: Daniel Rémy
First Print: 2007
© Société de formation à distance des commissions scolaires du Québec
All rights for translation and adaptation, in whole or in part, reserved for all
countries. Any reproduction, by mechanical or electronic means, including microre-
production, is forbidden without the written permission of a duly authorized
representative of the Société de formation distance des commissions scolaires du
Legal Deposit — 2007
Bibliothèque et Archives nationales du Québec
ISBN 978-2-89493-287-2
0.3
MTH-4109-1 Sets, Relations and Functions
Introduction to the Program Flowchart ................................................ 0.6
Program Flowchart................................................................................. 0.7
How to Use This Guide........................................................................... 0.8
General Introduction ..............................................................................0.11
Intermediate and Terminal Objectives of the Module.......................... 0.12
Diagnostic Test on the Prerequisites..................................................... 0.23
Answer Key for the Diagnostic Test on the Prerequisites ................... 0.27
Analysis of the Diagnostic Test Results ................................................ 0.29
Information for Distance Education Students ...................................... 0.31
UNITS
1. Sets of Numbers and Descriptions of Sets ............................................. 1.1
2. Set Relations: Inclusions and Equalities ............................................... 2.1
3. Operations and Series of Operations on Sets ........................................ 3.1
4. Set Operations Involving Real Numbers ............................................... 4.1
5. Cartesian Product.................................................................................... 5.1
6. Relations .................................................................................................. 6.1
7. Defining a Relation Using Set-Builder Notation ................................... 7.1
8. Functions ................................................................................................. 8.1
9. Cartesian Graph of a Function ............................................................... 9.1
10. Parabolas ............................................................................................... 10.1
11. Equations of a First- or Second-Degree Polynomial Function ............ 11.1
12. Describing the Characteristics of Various Functions .......................... 12.1
13. Comparative Analysis of Functional Situations .................................. 13.1
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MTH-4109-1 Sets, Relations and Functions
Final Review .......................................................................................... 14.1
Answer Key for the Final Review ........................................................ 14.26
Terminal Objectives ..............................................................................14.39
Self-Evaluation Test ..............................................................................14.41
Answer Key for the Self-Evaluation Test............................................. 14.59
Analysis of the Self-Evaluation Test Results....................................... 14.67
Final Evaluation....................................................................................14.68
Glossary ................................................................................................. 14.69
List of Symbols ...................................................................................... 14.75
Bibliography .......................................................................................... 14.76
Review Activities ................................................................................... 15.1
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MTH-4109-1 Sets, Relations and Functions
INTRODUCTION TO THE PROGRAM FLOWCHART
Welcome to the World of Mathematics!
This mathematics program has been developed for the adult students of the
Adult Education Services of school boards and distance education. The learning
activities have been designed for individualized learning. If you encounter
difficulties, do not hesitate to consult your teacher or to telephone the resource
person assigned to you. The following flowchart shows where this module fits
into the overall program. It allows you to see how far you have progressed and
how much you still have to do to achieve your vocational goal. There are several
possible paths you can take, depending on your chosen goal.
The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2
(MTH-416), and leads to a Diploma of Vocational Studies (DVS).
The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2
(MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School
Diploma (SSD), which allows you to enroll in certain Cegep-level programs that
do not call for a knowledge of advanced mathematics.
The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2
(MTH-536), and leads to Cegep programs that call for a solid knowledge of
mathematics in addition to other abiliies.
If this is your first contact with this mathematics program, consult the flowchart
on the next page and then read the section “How to Use This Guide.” Otherwise,
go directly to the section entitled “General Introduction.” Enjoy your work!
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MTH-4109-1 Sets, Relations and Functions
PROGRAM FLOWCHART
CEGEP
MTH-5110-1
Introduction to Vectors
MTH-5109-1
Geometry IV
MTH-5108-2
Trigonometric Functions and Equations
MTH-5107-2
Exponential and Logarithmic Functions
and Equations
MTH-5106-1
Real Functions and Equations
MTH-5105-1
Conics
MTH-5104-1
Optimization II
MTH-5103-1
Probability II
MTH-5102-1
Statistics III
MTH-5101-1
Optimization I
MTH-4110-1
The Four Operations on
Algebraic Fractions
MTH-4109-1
Sets, Relations and Functions
MTH-4108-1
MTH-4107-1
Straight Lines II
MTH-4106-1
Factoring and Algebraic Functions
MTH-4105-1
MTH-4103-1
Trigonometry I
MTH-4102-1
Geometry III
MTH-536
MTH-526
MTH-514
MTH-436
MTH-426
MTH-416
MTH-314
MTH-216
MTH-116
MTH-3002-2
Geometry II
MTH-3001-2
The Four Operations on Polynomials
MTH-2008-2
Statistics and Probabilities I
MTH-2007-2
Geometry I
MTH-2006-2
Equations and Inequalities I
MTH-1007-2
Decimals and Percent
MTH-1006-2
The Four Operations on Fractions
MTH-1005-2
The Four Operations on Integers
MTH-5111-2
Complement and Synthesis II
MTH-4111-2
Complement and Synthesis I
MTH-4101-2
Equations and Inequalities II
MTH-3003-2
Straight Lines I
DVS
MTH-5112-1
Logic
25 hours = 1 credit
50 hours = 2 credits
MTH-4104-2
Statistics II
You ar e h ere
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MTH-4109-1 Sets, Relations and Functions
Hi! My name is Monica and I have been
I’m Andy.
Whether you are
registered at an
center or pur-
suing distance
education, ...
You’ll see that with this method, math is
a real breeze!
... you have probably taken a
placement test which tells you
exactly which module you
My results on the test
indicate that I should begin
with this module.
Now, the module you have in your
hands is divided into three
sections. The first section is...
... the entry activity, which
contains the test on the
prerequisites.
By carefully correcting this test using the
ing your results on the analysis sheet ...
HOW TO USE THIS GUIDE
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MTH-4109-1 Sets, Relations and Functions
?
The memo pad signals a brief reminder of
concepts which you have already studied.
The calculator symbol reminds you that
you will need to use your calculator.
The sheaf of wheat indicates a review designed to
reinforce what you have just learned. A row of
sheaves near the end of the module indicates the
final review, which helps you to interrelate all the
learning activities in the module.
The starting line
shows where the
learning activities
begin.
The little white question mark indicates the questions
for which answers are given in the text.
?
... you can tell if you’re well enough
prepared to do all the activities in the
module.
The boldface question mark
indicates practice exercises
which allow you to try out what
you have just learned.
And if I’m not, if I need a little
review before moving on, what
happens then?
In that case, before you start the
activities in the module, the results
analysis chart refers you to a review
activity near the end of the module.
In this way, I can be sure I
have all the prerequisites
for starting.
Exactly! The second section
contains the learning activities. It’s
the main part of the module.
Look closely at the box to
the right. It explains the
symbols used to identify the
various activities.
The target precedes the
objective to be met.
Good!
?
START
Lastly, the finish line indicates
that it is time to go on to the self-evaluation
test to verify how well you have understood
the learning activities.
FINISH
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MTH-4109-1 Sets, Relations and Functions
A “Did you know that...”?
Later ...
For example, words in bold-
face italics appear in the
glossary at the end of the
module...
Great!
... statements in boxes are important
points to remember, like definitions, for-
mulas and rules. I’m telling you, the for-
mat makes everything much easier.
The third section contains the final re-
view, which interrelates the different
parts of the module.
Yes, for example, short tidbits
on the history of mathematics
and fun puzzles. They are in-
teresting and relieve tension at
the same time.
No, it’s not part of the learn-
ing activity. It’s just there to
give you a breather.
There are also many fun things
in this module. For example,
when you see the drawing of a
sage, it introduces a “Did you
know that...”
Must I memorize what the sage says?
It’s the same for the “math whiz”
pages, which are designed espe-
cially for those who love math.
And the whole module has
been arranged to make
learning easier.
There is also a self-evaluation
test and answer key. They tell
you if you’re ready for the final
evaluation.
Thanks, Monica, you’ve been a
big help.
I’ve got to run.
See you!
This is great! I never thought that I would
like mathematics as much as this!
They are so stimulating that
even if you don’t have to do
them, you’ll still want to.
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MTH-4109-1 Sets, Relations and Functions
GENERAL INTRODUCTION
SETS, RELATIONS AND FUNCTIONS
Every day, we are involved in some activity that requires us to match or group
items, make connections between individuals or objects, or make an informed
choice on the basis of a situational analysis. This is what we are going to examine
in this module, but applied to mathematics.
The MTH-4109-1 mathematics course is divided into three main sections:
sets;
relations;
functions.
The first part is aimed at helping you develop a thorough understanding of set
theory and the language used in this branch of mathematics.
The second part will allow you to master the concept of relation and the formal
language associated with it.
The study of these first two parts will introduce you to functions, the most
important part of this module. This third part will help you to master the
concepts relating to functions, to discover various functional situations, to
represent these situations and to analyze them.
This learning guide is written in clear and simple language, without sacrificing
mathematical rigour. Remember that at all times you are the main architect of
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MTH-4109-1 Sets, Relations and Functions
INTERMEDIATE AND TERMINAL OBJECTIVES OF
THE MODULE
Module MTH-4109-1 contains 25 objectives and requires 25 hours of study
distributed as shown below. The terminal objectives appear in boldface.
Objectives Number of hours* % (Evaluation)
1 to 6 2 10%
7 to 11 15%
12 and 13 15%
14 to 16 2 5%
17 1 5%
18 1 5%
19 2 5%
20 and 21 3 10%
22 3 10%
23 4 20%
24 and 25 4 20%
* One hour is allotted for the final evaluation.
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MTH-4109-1 Sets, Relations and Functions
1. Describing a set of integers by listing its elements, drawing a Venn diagram
or using set-builder notation
Describe a given finite or infinite set of integers ( ) by listing its elements,
using set-builder notation or drawing a Venn diagram. The set is determined
by an inequality of the form, x < n, x > n, n
1
< x < n
2
, x n, x n, n
1
x n
2
,
or that contains prime numbers, even numbers, odd numbers, squared
numbers, cubed numbers, multiples or factors of a given number. Given a set
whose elements are listed or described by means of set-builder notation or a
Venn diagram, convert the given description to one of the other descriptive
forms.
2. Relation of membership of an element to a set of numbers
Indicate whether or not an element belongs to a particular set by using the
appropriate symbol (i.e. x E if element x belongs to set E or x E if element
x does not belong to set E).
3. Relation of inclusion or equality between two sets
Determine whether there is a relation of inclusion or equality between two
sets. Express the relationship between each pair of sets by means of the
following symbols: if a given set is included within the other set; if a given
set is not included within the other set; = if the given sets are equal; if the
sets are not equal. Given a list of sets, indicate those that are the subsets of
a particular set. In most cases, the elements of these sets are listed.
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MTH-4109-1 Sets, Relations and Functions
4. Series of set operations on sets described by listing their elements
Given a universal set U, find the union ( ), intersection ( ) or difference (\)
of two given sets or find the complement (') of a particular set by correctly
applying the definitions below:
•A B = {x U|x A or x B},
•A B = {x U|x A and x B},
A \ B = {x U|x A and x B},
A' = {x U|x A}.
Sets A and B are finite or infinite sets whose elements are listed.
5. Describing a set of real numbers by drawing a graph or by using set-builder
or interval notation
Given a finite or infinite set of real numbers ( ) whose content is determined
by an inequality and given an interval of real numbers described by means
of set-builder notation, graph the set on the number line or indicate it using
the appropriate symbolic notation to be selected from the list below:
[a, b][a, b[]a, b]]a, b[
[a, ]a, , b]–, b[
In this case, a and b are the bounds of the interval. Given an interval written
in brackets, graph it on a number line and vice versa.
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MTH-4109-1 Sets, Relations and Functions
6. Series of set operations
Perform a series of two set operations ( , ,\,') on a maximum of three
intervals of real numbers that are written in square brackets,
graphed on a number line or described by means of set-builder
notation. The result of the operation or series of operations must be
written in square brackets, graphed on the number line or described
by means of set-builder notation.
7. Cartesian product of two sets
Find the Cartesian product of two sets whose elements are listed or defined
by means of set-builder notation or a Venn diagram by using the appropriate
symbols (i.e. A × B = {(x, y)|x A and y B}) to represent the Cartesian
product of set A by set B.
8. Distinguishing between source and target sets
Find the source and target sets that are the finite or infinite subsets of ,
or . By definition, the first set in a Cartesian product is the source set and
the second, the target set.
9. Subset of a Cartesian product
Given the Cartesian product of two sets and a rule of correspondence
indicating how the elements of these two sets are related, form a subset of this
Cartesian product with all the elements which make that sentence true by
using the appropriate symbols.
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MTH-4109-1 Sets, Relations and Functions
10. Source set, target set, domain and range of a relation, specifying the relation
of inclusion or equality
Find the source set, target set, domain and range that are finite or infinite
subsets of
, or . By definition, the set of the first coordinates of the
ordered pairs that are part of the solution set is called the domain and the
set of the second coordinates, the range. Determine whether a relation of
inclusion or equality exists between these sets.
11. Defining a relation using set-builder notation
Given a relation whose elements are listed or represented by a
graph, define it using set-builder notation. The rule of
correspondence must be expressed as a first- or second-degree
equation or inequality in one or two variables in
× .
Also determine the domain and the range of this relation.
12. Drawing the graph of an inequality
Draw the Cartesian graph in × of a first-degree inequality in one or two
variables.
13. Drawing the Cartesian graph of a relation defined by means of set-
builder notation
Draw the Cartesian graph of a relation whose elements are listed in
a subset of × . The rule of correspondence must be expressed as
a first-degree equation or inequality in one or two variables. Deter-
mine the domain and the range of this relation and define these sets
using set-builder notation.
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MTH-4109-1 Sets, Relations and Functions
14. Determining a function
For a concrete situation expressed in the form of a written statement, a
graph, a table of values or its rule, determine whether the relation is a
function, i.e., whether each element used in the source set corresponds to at
most one element in the target set.
15. Determining the dependent and independent variable in a functional
situation
Given a function described in set-builder notation, by listing its elements, by
means of a Venn diagram or a Cartesian graph, determine the dependent
variable, or the variable whose values are determined by the values of the
so-called independent variable.
16. Determining a function by means of functional notation
Determine the source set, the target set and the rule of correspondence and
represent these by using the apppropriate symbols of functional notation:
f :
x f(x), where f(x) is the rule of correspondence.
17. Extracting information
Extract information from the graph of various functions.
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MTH-4109-1 Sets, Relations and Functions
18. Describing the characteristics of a function
Given a functional situation represented by a Cartesian graph, describe the
characteristics of the function:
rate of change
type of variation
increasing or decreasing
domain and range
•sign
maximum values or minimum values, if any
x-intercept(s) (zeros)
y-intercept
axis of symmetry, if any
19. Drawing the graph of a parabola
Draw a parabola given an equation of the form y = ax
2
+ bx+ c or
y = a(xh)
2
+ k and determine the relation between the standard form and
the general form of the equation. The characteristics of each curve must be
indicated on the graph (vertex, zero(s), y-intercept).
20. Standard or general form of the equation of a parabola
Determine the standard or general form of the equation of a parabola given
the vertex and another of its points, or its zeros and another point. Using
the coordinates of the vertex and of another point or of the zeros and of
another point, determine the standard form (f (x) = a(xh)
2
+ k, a 0) and
the general form (f (x) = ax
2
+ bx + c, a 0) of the equation of a quadratic
function.
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MTH-4109-1 Sets, Relations and Functions
21. Determining the equation of a function given a written statement
Given a written statement providing relevant information about a
functional situation, find the equation of the function correspond-
ing to this situation, then write the equation in functional notation.
The written statement can be described by one of the following
functions:
a first-degree polynomial function, given two points or the slope
and a point;
a second-degree polynomial function, given the zeros and a
point, or the vertex and a point.
22. Translating from one mode of representation to another
Translate from one mode of functional representation to another according
to the possibilities listed in the table below.
FROM
TO
Words Table of
Graph
Rule or
values equation
Words (1)
Table of
(1)
values
Graph (1)
Rule or
(2)
equation
(1) When translating a written statement, a graph or a table of values into a rule
or an equation, students are limited to the following cases:
a first-degree polynomial function, given two points or the slope and a
point;
a second-degree polynomial function, given the zeros and a point, or the
vertex and a point.
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MTH-4109-1 Sets, Relations and Functions
(2) When translating a rule or an equation into a graph, various situations are
possible: polynomial functions, inverse-variation functions, rational func-
tions, square-root functions, greatest-integer functions, absolute-value
functions, exponential functions, etc. A technological tool may be used in the
case of a function that has not been covered in class or in previous courses.
23. Drawing the graph of a functional situation
Given a functional situation described in the form of a written
statement, a table of values or a rule, draw its corresponding
Cartesian graph and describe the characteristics of the function.
24. Determining the values that make up the domain or the range in a
functional situation
Determine or estimate certain values that make up the domain or
the range in a functional situation described in the form of a written
statement, a Cartesian graph or a rule. The situation may be
described by a combination of two or more functions over
consecutive intervals.
25. Comparative analysis of functional situations
Solve problems by performing a comparative analysis of similar
functional situations. Each situation must be described as a
function presented in the form of a written statement, a table of
values, a rule or a graph.
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MTH-4109-1 Sets, Relations and Functions
The 25 objectives in this module are covered in 13 units, as outlined below.
Unit Objective(s)
1 Describing a set of integers 1
Relation of membership 2
2 Relation of inclusion or equality 3
3 Series of set operations on sets described by listing their elements 4
4 Describing a set of real numbers 5
Series of set operations 6
5 Cartesian product of two sets 7
Distinguishing between source and target sets 8
6 Subset of a Cartesian product 9
Source set, target set, domain and range of a relation, specifying the
relation of inclusion or equality 10
Defining a relation using set-builder notation 11
7 Drawing the graph of an inequality 12
Drawing the Cartesian graph of a relation
defined by means of set-builder notation 13
8 Determining a function 14
Determining the dependent and the independent variable 15
Determining a function by means of functional notation 16
9 Extracting information 17
Describing the characteristics of a function 18
10 Drawing the graph of a parabola 19
Standard or general form of the equation of a parabola 20
11 Determining the equation of a function given a written statement 21
12 Translating from one mode of representation to another 22
Drawing the graph of a functional situation 23
13 Determining the values that make up the domain or the range
in a functional situation 24
Comparative analysis of functional situations 25
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MTH-4109-1 Sets, Relations and Functions
DIAGNOSTIC TEST ON THE PREREQUISTES
Instructions
1. Answer as many questions as you can.
2. You may use a calculator.
4. Don’t waste any time. If you cannot answer a question, go on to
the next one immediately.
5. When you have answered as many questions as you can, correct
test.
those in the key. In addition, the various steps in your solution
should be equivalent to those shown in the answer key.
key. This chart gives an analysis of the diagnostic test results.
8. Do the review activities that apply to each of your incorrect
9. If all your answers are correct, you may begin working on
this module.
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MTH-4109-1 Sets, Relations and Functions
1
y
x
E
D
F
C
B
01
A
1. Indicate:
a) all the prime numbers less than 30. ..........................................................
b) all the factors or divisors of 30. ..................................................................
c) all the multiples of 6. ..................................................................................
d) all the divisors of 50. ...................................................................................
e) all the even prime numbers........................................................................
2. Note the following number line.
0
ABC DE
F
•• •••
4
a) What is the coordinate of each of the following points?
A: ................ B: ................ C: ................ D: .................
b) Identify the point whose coordinate is 6. ...................................................
c) Plot point H on the above number line if its coordinate is –8.
3. Note the Cartesian coordinate
system on the right:
a) Determine the coordinates of the following points.
A: ................. B: ................. C: ..................
D: ................. E: ................. F: ..................
b) What is the x-coordinate of point A? ..........................................................
c) What is the y-coordinate of point C? ..........................................................
d) Name a point in Quadrant II. ....................................................................
e) Name a point located on the x-axis. ...........................................................
f) Give a synonym for "y-axis.".......................................................................
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MTH-4109-1 Sets, Relations and Functions
••
1
y
x
C
D
0
1
A
B
4. Note the Cartesian coordinate system on the right:
Determine the equation of straight
line AB and straight line CD and
write them in the form y = mx + b.
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MTH-4109-1 Sets, Relations and Functions
ANSWER KEY FOR THE DIAGNOSTIC TEST
ON THE PREREQUISITES
1. a) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
b) 1, 2, 3, 5, 6, 10, 15, 30
c) 0, 6, 12, 18, 24, 30, …
d) 1, 2, 5, 10, 25, 50
e) 2
2. a) A: –10 B: –6 C: –4 D: 2
b) E
c)
0
ABC DE
F
•• •••
–10 –8 –6 –4 –2 2 4 6 8 10
H
3. a) A(–2, 3), B(0, 3), C(1, 2), D(–2, 0), E(–4, –1), F(2, –1).
b) 2 c) 2 d) A e) D
f) Axis of ordinates
4. a) Equation of straight line AB
Given points A(3, 0) and B(0, 3), calculate the slope.
m =
y
2
y
1
x
2
x
1
=
3–0
0–3
=
3
–3
= –1
Since m = –1 and b = 3 (y-intercept), it follows that y = –1x + 3.
The equation of straight line AB is therefore y = –x + 3.
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MTH-4109-1 Sets, Relations and Functions
b) Equation of straight line CD
Given points C(–2, –1) and D(1, 3), calculate the slope.
m =
y
2
y
1
x
2
x
1
=
3–(1)
1–(2)
=
3+1
1+2
=
4
3
Hence,
4
3
=
y –3
x –1
3(y – 3) = 4(x – 1)
3y – 9 = 4x – 4
3y = 4x – 4 + 9
3y = 4x + 5
y =
4
3
x +
5
3
The equation of straight line CD is therefore y =
4
3
x +
5
3
.
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MTH-4109-1 Sets, Relations and Functions
ANALYSIS OF THE DIAGNOSTIC
TEST RESULTS
Questions
Correct Incorrect Section Page Unit(s)
1. a) 15.1 15.4 Unit 1
b) 15.1 15.4 Unit 1
c) 15.1 15.4 Unit 1
d) 15.1 15.4 Unit 1
e) 15.1 15.4 Unit 1
2. a) 15.2 15.11 Unit 4
b) 15.2 15.11 Unit 4
c) 15.2 15.11 Unit 4
3. a) 15.3 15.14 Unit 5
b) 15.3 15.14 Unit 5
c) 15.3 15.14 Unit 5
d) 15.3 15.14 Unit 5
e) 15.3 15.14 Unit 5
f) 15.3 15.14 Unit 5
4. a) 15.3 15.14 Unit 6
b) 15.3 15.14 Unit 6
If all of your answers are correct, you may begin working on this module.
For each incorrect answer, find the related section listed in the Review
column. Do the review activities for that section before beginning the units
listed in the right-hand column under the heading Before Going to Unit(s).
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0.31
MTH-4109-1 Sets, Relations and Functions
INFORMATION FOR DISTANCE
EDUCATION STUDENTS
You now have the learning material for MTH-4109-1 and the relevant homework
assignments. Enclosed with this package is a letter of introduction from your
tutor, indicating the various ways in which you can communicate with him or her
(e.g. by letter or telephone), as well as the times when he or she is available. Your
make use of his or her services if you have any questions.
DEVELOPING EFFECTIVE STUDY HABITS
Learning by correspondence is a process which offers considerable flexibility, but
which also requires active involvement on your part. It demands regular study
and sustained effort. Efficient study habits will simplify your task. To ensure
effective and continuous progress in your studies, it is strongly recommended
that you:
draw up a study timetable that takes your work habits into account and is
compatible with your leisure and other activities;
develop a habit of regular and concentrated study.
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0.32
MTH-4109-1 Sets, Relations and Functions
The following guidelines concerning theory, examples, exercises and assign-
Theory
To make sure you grasp the theoretical concepts thoroughly:
1. Read the lesson carefully and underline the important points.
2. Memorize the definitions, formulas and procedures used to solve a given
problem; this will make the lesson much easier to understand.
3. At the end of the homework assignment, make a note of any points that you
do not understand using the sheets provided for this purpose. Your tutor will
then be able to give you pertinent explanations.
4. Try to continue studying even if you run into a problem. However, if a major
homework assignment, using the procedures outlined in the letter of
introduction.
Examples
The examples given throughout the course are applications of the theory you are
studying. They illustrate the steps involved in doing the exercises. Carefully
study the solutions given in the examples and redo the examples yourself before
starting the exercises.
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0.33
MTH-4109-1 Sets, Relations and Functions
Exercises
The exercises in each unit are generally modeled on the examples provided. Here
1. Write up your solutions, using the examples in the unit as models. It is
important not to refer to the answer key found on the coloured pages at the
back of the module until you have completed the exercises.
2. Compare your solutions with those in the answer key only after having done
all the exercises. Careful! Examine the steps in your solutions carefully,
3. If you find a mistake in your answer or solution, review the concepts that you
did not understand, as well as the pertinent examples. Then redo the
exercise.
4. Make sure you have successfully completed all the exercises in a unit before
moving on to the next one.
Homework Assignments
Module MTH-4109-1 contains three homework assignments. The first page of
each assignment indicates the units to which the questions refer. The homework
assignments are designed to evaluate how well you have understood the
material studied. They also provide a means of communicating with your tutor.
When you have understood the material and have successfully completed the
pertinent exercises, do the corresponding assignment right away. Here are a few
suggestions:
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0.34
MTH-4109-1 Sets, Relations and Functions
1. Do a rough draft first, and then, if necessary, revise your solutions before
2. Copy out your final answers or solutions in the blank spaces of the document
to be sent to your tutor. It is best to use a pencil.
3. Include a clear and detailed solution with the answer if the problem involves
several steps.
4. Mail only one homework assignment at a time. After correcting the assign-
ment, your tutor will return it to you.
In the section “Student’s Questions,” write any questions which you wish to have
studies, if necessary.
In this course
Homework Assignment 1 is based on units 1 to 7.
Homework Assignment 2 is based on units 8 to 13.
Homework Assignment 3 is based on units 1 to 13.
CERTIFICATION
When you have completed all your work, and provided you have maintained an
average of at least 60%, you will be eligible to write the examination for this
course.
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1.1
MTH-4109-1 Sets, Relations and Functions
UNIT 1
SETS OF NUMBERS AND
DESCRIPTIONS OF SETS
1.1 SETTING THE CONTEXT
Sets and Families
Anthony is trying to help his friends understand the concept of a set. He takes
them to the largest record store in his neighbourhood and shows them how the
various recordings are divided into different sections.
Anthony explains that all the recording categories are part of a large set. Within
this set, there are subcategories, such as pop rock, rap, jazz and classical. These
are also sets. Other sets are possible, since the pop rock category may contain
French pop and English pop.
His friends have understood that a set is a collection of objects or group of people
with common characteristics. Anthony could have taken his friends to any store
or visited any Web site to help them understand the concept of a set.
START
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1.2
MTH-4109-1 Sets, Relations and Functions
To achieve the objective of this unit, you should be able to indicate
whether or not a given element belongs to a set of numbers. In addition,
you should be able to describe a set of integers by listing its elements, by
using set-builder notation or by drawing a Venn diagram.
In everyday life and in mathematics as well, the term "set" is often used. A set
contains objects, people or numbers with something in common and forms a well-
defined whole. Each object included in the set is called an "element" of that set.
A set is a clearly defined collection of separate numbers,
objects or people, who usually share one or more common
characteristics, called "elements of the set."
When elements share a particular characteristic, we can say that they are part
of a certain set. This characteristic can be as simple as a colour, a family tie, a
group of numbers, and so on.
There are three different methods of representing a set:
1. by listing its elements;
2. by using set-builder notation;
3. by drawing a Venn diagram.
The method you choose will depend on how it will be used to solve the problem
in question.
Describing a Set by Listing Its Elements
We can describe a set by writing down the elements of that set one after another
without repetition. When we do this, we are "describing a set by listing its
elements."
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1.3
MTH-4109-1 Sets, Relations and Functions
To describe a set by listing its elements is to indicate in
brackets the elements belonging to that set without repeating
any element. An uppercase letter is used to identify the set.
Example 1
If set A is the set of natural numbers less than 5, describe it by listing its
elements.
This is written as follows: A = {0, 1, 2, 3, 4}.
Note that none of the elements has been omitted and that each element has
the required characteristics.
Rules to follow when describing a set by listing its
elements:
1. Use an uppercase letter to identify the set.
2. Write the symbol = after the uppercase letter.
3. Put the elements in brackets and separate them with
commas.
4. Name each element only once.
5. Note that the order of the elements is not important.
However, to make it easier to read sets containing num-
bers, it is preferable to put them in ascending or descend-
ing order.
6. Use suspension points (...) when you cannot list all the
elements of a set. These points are preceded by a comma
and replace the missing elements.
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1.4
MTH-4109-1 Sets, Relations and Functions
Here are some examples of describing a set by listing its elements.
Examples 2
a) List the elements of the set of even natural numbers that are less
than 10.
If we call this set A, then A = {0, 2, 4, 6, 8}.
Note that the elements of this set have the characteristics required for this
set: the elements are natural numbers that are even and less than 10.
Each element in this set corresponds to this definition: no element has
been omitted and no element lacking these characteristics has been
included.
b) Now describe the set of natural numbers that are multiples of 4.
The result is: M = {0, 4, 8, 12, 16, 20, ...}.
This time, we used suspension points because the set does not have a finite
number of elements. In such cases, we list 5 or 6 elements that make the
set in question easier to identify.
If, for instance, we had written M = {0, 4, ...}, the reader might have
thought that this represented the set of squares of even natural numbers.
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1.5
MTH-4109-1 Sets, Relations and Functions
: the set of natural numbers
= {0, 1, 2, 3, 4, 5, ....}
: the set of integers
= {..., –3, –2, – 1, 0, 1, 2, 3, ...}
: the set of rational numbers
= {
..., 50, ..., 4,2, ...,
1
2
, ..., 0, ..., 1, ...,
15
4
, ...
}
Exercise 1.1
1. List the elements of each of the following sets.
a) The odd natural numbers less than 16.
Set A = .........................................................................................................
b) The squares of the first 5 even natural numbers.
Set B = .........................................................................................................
c) The integers between –6 and 6.
Set C = .........................................................................................................
d) The integers less than 2.
Set D = .........................................................................................................
e) The integers greater than –2.
Set E = .........................................................................................................
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1.6
MTH-4109-1 Sets, Relations and Functions
A finite set is a set with a limited number of elements. This
number is called the cardinal number of a set.
An infinite set is a set with an infinite number of elements.
Example 3
a) If set A is the set of integers less than or equal to 5, describe it by listing
its elements.
We get: A = {..., –2, –1, 0, 1, 2, 3, 4, 5}.
b) If set B is the set of integers greater than or equal to 5, describe it by listing
its elements.
We get: B = {5, 6, 7, 8, 9, 10, 11, ...}.
c) If set C is the set of even integers between –200 and 200, describe it by
listing its elements.
We get: C = {–198, –196, –194, ..., 194, 196, 198}.
Note that in this example, all the sets whose elements are listed include three
suspension points, but that only sets A and B are infinite. Set C was shortened
because there were too many elements. We can easily identify the set by listing
the first and the last three elements of the set.
The cardinal number of a set is the number of elements
that belong to the set. The cardinal number of set E is denoted
by n(E).
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1.7
MTH-4109-1 Sets, Relations and Functions
For example, set A = {0, 2, 4, 6, 8} is a finite set and n(A) = 5, whereas
M = {0, 4, 8, 12, 16, 20, ...} is an infinite set because we cannot determine the exact
number of elements it contains.
? What is the cardinal number of a set with no elements? ..............................
The cardinal number of such a set is 0 and it is called an "empty set."
An empty set is a set without any elements. It is denoted by
{ } or Ø, which is read "phi."
If the elements of a set have already been listed, we can describe it in words. In
this case, we should identify as many characteristics of the set as possible so that
it won't be confused with another.
Example 4
Let set A = {16, 20, 24, 28, 32, 36, ...}.
In examining the elements of this set, we can see that:
1. they are all natural numbers;
2. they are all multiples of 4;
3. these multiples of 4 are all greater than or equal to 16.
We can conclude that these elements represent the following set: natural
numbers that are multiples of 4 and greater than or equal to 16.
Let B = {0, 1, 4, 9, 16, 25, 36, ...}.
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1.8
MTH-4109-1 Sets, Relations and Functions
? What can we say about the elements of set B?
...........................................................................................................................
? Use words to describe this set as accurately as possible.
...........................................................................................................................
The elements of B are all squares. Hence, we can say that set B is the set of
natural numbers that are squares.
Exercise 1.2
1. List the elements of each set below. State whether the set is finite or infinite
and give its cardinal number, if applicable.
a) The odd natural numbers.
A =............................................... ....................... ........................
b) The natural numbers that are multiples of 5 and less than 200.
B =............................................... ....................... ........................
c) The prime numbers less than 50.
C =............................................... ....................... ........................
d) The squares of the first 8 natural numbers.
D = .............................................. ....................... ........................
e) The divisors of 24.
E =............................................... ....................... ........................
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1.9
MTH-4109-1 Sets, Relations and Functions
f) The integers less than 4 and greater than –3.
F = ............................................... ....................... ........................
g) The cubes of the first 6 natural numbers.
G = .............................................. ....................... ........................
h) The natural numbers less than 0.
H = .............................................. ....................... ........................
2. Give the characteristics of each of the following sets.
a) A = {0, 1, 2, 3, ..., 98, 99, 100}
.......................................................................................................................
b) B = {0, 8, 16, 24, 32, ...}
.......................................................................................................................
c) C = {..., –2, –1, 0, 1, 2, 3}
.......................................................................................................................
d) D = {1, 2, 3, 4, 6, 12}
.......................................................................................................................
e) E = {1, 8, 27, 64}
.......................................................................................................................
f) F = {3, 4, 5, 6}
.......................................................................................................................
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1.10
MTH-4109-1 Sets, Relations and Functions
g) G = {..., –3, –2, –1, 0, 1, 2, 3, ...}
.......................................................................................................................
h) H = {0, 2, 4, ..., 94, 96, 98}
.......................................................................................................................
of certain sets. Correct the errors.
a) A = {1 2 3 4 5} ...................................................................
b) B = {8, 8, 9, 10, 11, 11, 12} ...................................................................
c) c = {13, 14, 15, 16, 17, ...} ...................................................................
d) D = 1, 3, 5, 7, 9, 11 ...................................................................
We will know that a set is clearly defined if we can determine whether or not any
given element belongs to that set.
To indicate that any element belongs to a set, we use the symbol for
membership, which is denoted by . For instance, the expression x C means
that element x is part of set C, that it is an element of set C or that it belongs to
set C.
We know that the element 4 belongs to the set of even natural numbers
P = {0, 2, 4, 6, 8, ...}. Hence, we can write 4 P.
To indicate that an element x belongs to a given set A, we
write x A.
x A means "x is an element of set A"
or
"x belongs to set A."
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1.11
MTH-4109-1 Sets, Relations and Functions
When an element does not belong to a given set, we use the symbol . If set E
is made up of even natural numbers, then the number 7 does not belong to it.
Hence, we write 7 E.
To indicate that an element x does not belong to a given
set A, we write x A.
x A means "x is not an element of set A"
or
"x does not belong to set A."
Example 5
Let A = {–3, –2, –1, 0, 1}.
We can write –3 A, –2 A, –1 A, 0 A, 1 A.
–3 A reads: –3 is an element of set A
or
3 belongs to set A.
We can also write 3 A, 5 A, 7 A.
3 A reads: 3 is not an element of set A
or
3 does not belong to set A.
We can also use the symbol for any other numbers, except those that are
part of set A.
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1.12
MTH-4109-1 Sets, Relations and Functions
Procedure for determining if an element x belongs to a
given set A:
1. Look for the elements that belong to set A and list them.
2. Check if element x belongs to set A.
3. Write either x A or x A, as the case may be.
Example 6
Set A is the set of divisors of 12. Determine the truth value of the following
propositions 2 A, 8 A.
1. A = {1, 2, 3, 4, 6, 12}
2. Element 2 belongs to set A, but element 8 does not.
3. We can therefore write: 2 A and 8 A.
Exercise 1.3
1. If A is the set of all the divisors of 40, state whether the following statements
are true or false.
a) 2 A .................... b) 0 A .................... c) 80 A ......................
d) 1 A .................... e)
1
2
A .................... f) a A ......................
2. Given A = {1, 2, 8, 9} and B = {2, 4, 7, 8}, fill in the blanks by using either
or .
a) 2 ...... A b) 1 ...... A c) 4 ...... A
d) 2 ...... B e) 8 ...... B f) 9 ...... B
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1.13
MTH-4109-1 Sets, Relations and Functions
3. Using the appropriate symbol, indicate whether the following elements
belong to set A = {–50, 5, 10, 50}.
a) 50 ...... A b) 10 ...... A c) 1 ...... A
d) 10 ...... A e) 05 ...... A f) 0 ...... A
Describing a Set by Means of Set-Builder Notation
A second way of describing a set is to state one or more characteristics possessed
by all the elements of this set and by no element that does not. When we do this,
we are describing a set by means of set-builder notation.
The characteristics common to all the elements of the set must be specific enough
so that only one interpretation is possible. We should then be able to identify
these elements, and only these elements without having to list them. This
representation is more concise.
To describe a set using set-builder notation, first indi-
cate to which set of numbers the elements belong (universe)
and then indicate the property or properties shared by all the
elements of that set. This description is placed between
brackets and is preceded by the name of the set identified by
an uppercase letter.
Since we need not list any other elements belonging to the set, a lowercase letter
(usually x, y or z) will be used to designate all the elements of the set.
Let's look at an example of how to describe a set by means of set-builder notation.
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1.14
MTH-4109-1 Sets, Relations and Functions
Example 7
The elements of set A are listed as follows: A = {..., –9, –6, –3, 0, 3, 6, 9, ...}.
Describe this set by means of set-builder notation.
1. First, let x represent the elements of the set.
2. Since the elements of this set belong to set
, it becomes the universe and
we can write:
x
3. Since each element of this set is a multiple of 3, we write:
x is a multiple of 3.
4. These two mathematical expressions can be linked by the vertical line (|)
which means "such that."
5. We then write the name of the set and put the two mathematical
expressions in brackets.
The result is A = {x
|x is a multiple of 3}.
This mathematical expression means that A is the set of all the x elements
belonging to the set of integers such that x is a multiple of 3.
We have seen that when we describe a set using set-builder notation, we can use
mathematical language and symbols to state all the characteristics of the set
without listing a single element of the set.
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1.15
MTH-4109-1 Sets, Relations and Functions
Procedure for describing a set by means of set-builder
notation:
1. Choose a variable to represent the elements of the set.
2. Identify the universe using the symbol for membership to
show that the variable belongs to that universe
(e.g. x
).
3. Find the characteristic(s) the elements have in common
and express this in mathematical language.
4. Link the two mathematical expressions together using
the symbol | (such that).
5. Write the name of the set and then put the two mathemati-
cal expressions in brackets.
To express order relations between numbers, we use the following
mathematical symbols:
< for "less than";
> for "greater than";
= for "equal to";
for "less than or equal to";
for "greater than or equal to".
Example 8
Let B = {7, 8, 9, 10, 11}.
The characteristics of the elements of B are:
they are natural numbers: x ;
the numbers are between 6 and 12 exclusively, hence 6 < x < 12.
We write B = {x |6 < x < 12}.
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1.16
MTH-4109-1 Sets, Relations and Functions
The expression means that B is the set of all the x elements belonging to the
set of natural numbers such that x is between 6 and 12.
Note that the following are equivalent ways of describing this set by means
of set-builder notation.
B= {x
|7 x 11},
or B = {x
|6 < x 11},
or B = {x
|7 x < 12}.
To describe all the natural numbers between 6 and 12, we can
write 6 < x < 12. These numbers are 7, 8, 9, 10 and 11 because
x cannot be equal to 6 or 12. We can also write 7
x
11, which
means that x can be equal to 8, 9 or 10. Furthermore, the equals
sign (the line under the inequality sign) indicates that x can also
be equal to 7 or 11, so the value of x can be 7, 8, 9, 10 or 11. We
can also combine these two notations and obtain: 6 < x
11 or
7
x < 12.
These set-builder notations are not equivalent if we use a different universe,
such as
.
? Why are sets A and B not equivalent if:
A = {x |3 < x < 6} and B = {x |4 x 5}?
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
Set A contains all the elements of
that are greater than 3 and less than 6. On
the other hand, B contains all the elements of from 4 to 5. Hence, there are
rational elements missing from B if the two sets are supposed to contain the same
elements. For instance, 3.5 A but 3.5 B.
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1.17
MTH-4109-1 Sets, Relations and Functions
We have to be careful when describing a set using set-builder notation: it is
important to know the universe and to use the symbols <, >, and correctly.
The next step is to do the opposite of what we have just seen: given a set described
by means of set-builder notation, we will define the set by listing its elements.
Example 9
Let set C = {x
|x is a prime number that is a divisor of 12}. Describe this
set by listing its elements.
The description using set-builder notation tells us that:
the elements are natural numbers;
the elements are prime numbers;
the elements are divisors of 12.
The elements with these characteristics are 2 and 3.
Hence, the result is C = {2, 3}.
A prime number has exactly two divisors: 1 and itself. One is not
a prime number since its only divisor is 1.
When the elements of the set we are describing by means of set-builder notation
have more than one characteristic, we can indicate this by using the conjunction
and, which means that the elements must satisfy all the characteristics listed.
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1.18
MTH-4109-1 Sets, Relations and Functions
Example 10
D = {x
|x is an even number and x > –12}
We are looking for integers. These integers are even numbers and greater
than –12.
D = {–10, –8, –6, –4, ...}
? If D = {x |x is an even number and x > –12}, do we obtain the same set?
...........................................................................................................................
No, because the elements of D have to be natural numbers. If we list the elements
of D, we obtain D = {0, 2, 4, 6, 8, 10, ...}.
In some cases, there will be no elements to list, because no number will have the
characteristics described by means of set-builder notation. Study the following
example.
Example 11
List the elements of A = {x
|x < –5}.
Since no element of is negative, no element can be less than –5. Hence
A = { } or Ø.
Time for a few exercises!
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1.19
MTH-4109-1 Sets, Relations and Functions
Exercise 1.4
1. Use everyday language to describe the following sets given in set-builder
notation.
a) A = {x
|x > –3} .......................................................................................
.......................................................................................................................
b) B = {x
|x < 7 and x is an odd number} .................................................
.......................................................................................................................
c) C = {x
|x is a divisor of 0} .....................................................................
.......................................................................................................................
2. Describe the following sets by using set-builder notation.
a) A = {1, 3, 5, 7, 9, ...} ...............................................................
b) B = {0, 1, 2, 3} ...............................................................
c) C = {–27, –8, –1, 0, 1, 8, 27} ...............................................................
d) D = {1, 3, 5, 15} ...............................................................
e) E = {..., 0, 7, 14, 21, 28, ...} ...............................................................
3. Describe the following sets by listing their elements.
a) D = {x
|x is an even number}................................................................
b) E = {x |x 2} .......................................................................................
c) F = {x |x is a multiple of 100}...............................................................
d) G = {x |x is a divisor of 50} ....................................................................
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1.20
MTH-4109-1 Sets, Relations and Functions
e) J = {x |–2 < x < 6} .................................................................................
f) K = {x |–10 < x 6} ............................................................................
g) L = {x |x > 50 and x is a cube}..............................................................
h) M = {x |x < 8 and x is negative} ...........................................................
i) N = {x |x > 1/2} ......................................................................................
j) P = {x |5.35 < x < 6.73} .........................................................................
Describing a Set by Drawing a Venn Diagram
The third way of describing a set is to draw a closed figure (usually a circle)
containing points that represent all the elements belonging to the set. When we
do this, we are describing a set using a Venn diagram.
This technique is most often used to represent finite sets or sets of numbers that
we are very familiar with, such as natural numbers, integers and rational
numbers.
Later in this course, we will see that this technique can be used to solve many
problems related to sets.
To describe a set using a Venn diagram is to draw a
closed figure containing the points that represent all the
elements of the set.
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1.21
MTH-4109-1 Sets, Relations and Functions
Example 12
Describe the following set using a Venn diagram:
A = {x
|–4 < x < 6 and x is an even number}.
1. List the elements of the set: A = {–2, 0, 2, 4}.
2. Draw a circle containing a point for each
element listed.
3. Write the name of the set alongside the
circle.
Fig. 1.1 Diagram of set A
That's all there is to it! We now have a Venn diagram of set A, and we can easily
list its elements by simply naming the elements shown in the circle.
Most of the problems we will encounter will involve more than one set.
Let A = {0, 1, 2, 3} and B = {3, 4, 5}.
If the universe U = {0, 1, 2, 3, 4, 5, 6, 7}, we can describe these three sets using
a Venn diagram. The universe is represented by a rectangle. Since sets A and
B and the elements of these sets belong to the universe U, they will also be shown
in the rectangle.
•0
•2
•4
•–2
•0
•2
•4
•–2
A
1
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1.22
MTH-4109-1 Sets, Relations and Functions
•0
•1
•2
•4
•5
•6
•7
•3
AB
U
Fig. 1.2 Diagram of U, A and B
•3 A and 3 B: hence 3 goes in A and B.
•0 A, 1 A and 2 A: 0, 1 and 2 go in A only.
•4 B and 5 B: 4 and 5 go in B only.
•6 U and 7 U: 6 and 7 go in U only.
Example 13
Let A = {x
|2 < x < 7} and B = {x |x = 6}.
Describe these sets using a Venn diagram.
1. Describe sets A and B by listing their elements.
A = {3, 4, 5, 6} and B = {6}.
2. Draw a Venn diagram showing that is the universe.
AB
Fig. 1.3 Venn diagram
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1.23
MTH-4109-1 Sets, Relations and Functions
3. Indicate the elements of each set, beginning with those common to both
sets. Here, 6 is the common element. Then add the elements missing from
A, namely the numbers 3, 4 and 5. There is no element to add to B.
•1
•2
•7
•6
•4
A
B
•0
•3
•5
•8
.
.
.
Fig. 1.4 Diagram of A, B and
The elements 0, 1, 2, 7 and 8 as well as suspension points have been
inserted to complete the set . These additions are not really necessary
because we know the definition of . They will be omitted from now on.
Venn diagrams of three sets A, B and C belonging to the same universe U are also
quite common. Here is a diagram of this type of situation.
•1
•2
•7
U
•6
•4
A
B
•3
•5
•8
C
Fig. 1.5 Diagram of U, A, B and C
? Describe this universe by listing its elements. ...............................................
? Describe sets A, B and C by listing their elements. .......................................
...........................................................................................................................
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1.24
MTH-4109-1 Sets, Relations and Functions
State whether the following statements are true or false.
? a) 1 A ................. b) 9 U ................ c) 8 B .................
? d) 3 C ................. e) 1 U ................ f) 2 A .................
The universe is U = {1, 2, 3, 4, 5, 6, 7, 8}. Set A = {1, 2, 3, 5}, set B = {3, 4, 5, 8}
and set C = {5, 6, 8}. As for the statements, only a) is true; the others are all false.
Procedure for describing three sets using a Venn
diagram:
1. First, list all the elements.
2. Identify the elements that are common to all three sets.
3. Identify the elements that are common to only two sets.
4. Complete the diagram by adding the elements that are
missing from each set.
Exercise 1.5
1. Describe the following sets using a Venn diagram.
a) A = {0, 1, 4, 5, 8} b) B = {1, 4, 6, 8}
1
2
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1.25
MTH-4109-1 Sets, Relations and Functions
c) C = {x |–2 < x < 3 and x is even} d) D = {x |2 < x < 4}
2. Describe the sets shown in the Venn diagrams below, first by listing their
elements and then by means of set-builder notation.
a) Listing of elements
...........................................................
Set-builder notation
...........................................................
b) Listing of elements
...........................................................
Set-builder notation
...........................................................
c) Listing of elements
...........................................................
Set-builder notation
...........................................................
•7
A
•3
•5
•9
•11
•13
B
•14
•0
•7
•35
•28
•21
•1
•4
•0
C
•9
•36
•16
•25
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1.26
MTH-4109-1 Sets, Relations and Functions
d) Listing of elements
...........................................................
Set-builder notation
...........................................................
3. Note the Venn diagram on the right.
•1
•2
•0
•3
•5
A
B
•15
a) Describe A by listing its elements........................................
b) Describe B by listing its elements........................................
c) Describe A using set-builder notation. ...........................................
d) Describe B using set-builder notation. ...........................................
e) Fill in the blanks below with the symbols or .
2 ....... A 5 ....... B 1 ....... 3 ....... B
4. Let A = {x |3 < x < 8},
B = {x |x 12 and x is a multiple of 4},
C = {x |x is a divisor of 8}.
a) Describe sets A, B and C by listing their elements.
A = ........................... B = ............................ C = ...............................
•1
•2
•–4
•0
D
•–1
•–3
•–2
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1.27
MTH-4109-1 Sets, Relations and Functions
b) Describe these sets by completing the following Venn diagram.
A
B
C
5. Alicia decided to give her friend money as a birthday gift. She has \$25 (three
\$5 bills and one \$10 bill) and has no way to break the bills.
a) Describe the following sets by listing their elements.
U is the universe of all the different amounts Alicia can give her friend.
U = ..........................................................................................................
A is the set of the possible amounts Alicia can give her friend if she
decides to give her an uneven sum of money.
A = ..........................................................................................................
B is the set of the possible amounts Alicia can give her friend if she
decides to give her an even sum of money.
B = ..........................................................................................................
C is the set of the possible amounts Alicia can give her friend if she
gives her only two bills.
C = ..........................................................................................................
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1.28
MTH-4109-1 Sets, Relations and Functions
b) Represent all of these sets in the Venn diagram below.
U
A
B
C
Sets of Numbers: , , , ' and
Up to now, we have seen several sets of numbers and a brief description of each.
We are now going to summarize all the sets of numbers that belong to the set of
real numbers. We will also learn how to determine whether or not a number
belongs to each of these sets, how to draw a Venn diagram of them and how to
represent real numbers in these diagrams.
Natural Numbers:
When we count objects, for instance, we say that we have 0, 1, 2, 3, ... objects.
These numbers make up the set
.
The set is the set containing all the positive integers and
the number 0.
= {0, 1, 2, 3, ...}
* = {1, 2, 3, ...}
The asterisk indicates that zero does not belong to the set.
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1.29
MTH-4109-1 Sets, Relations and Functions
Integers:
Set is insufficient for certain subtractions. For instance, the result of 8 – 15
does not belong to , so we have to create a new set containing both positive and
negative integers. This new set is .
The set is the set of integers.
= {..., –2, –1, 0, 1, 2, ...}
Set
includes , as shown in the Venn diagram on the
right.
Rational Numbers:
Set is insufficient for certain types of divisions. For instance, the result of
8 ÷ 3 does not belong to , so we have to create a new set containing not only
integers, but also numbers expressed as fractions whose numerators are inte-
gers and whose denominators are integers other than 0. This new set is .
N.B. Division by 0 is impossible.
The set is the set of the numbers expressed as a quotient
of two integers. The second integer (the denominator) cannot
be 0.
=
{
a
b
|a , b *
}
N.B. We describe by means of set-builder notation because it would be
extemely difficult to list its elements.
Fig. 1.6 Venn diagram of and
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1.30
MTH-4109-1 Sets, Relations and Functions
The set includes and includes , as shown in the
Venn diagram on the right.
Rational numbers possess a very important property. To discover this property,
let's look at some rational numbers written in decimal form.
1
3
= 0.333... and is written 0.3;
4
11
= 0.363 636... and is written 0.36;
3 = 3.000... and is written 3.0;
1
4
= 0.25 = 0.250 00... and is written 0.250;
–2
1
7
= –2.142 857 142 857... and is written –2.142 857.
The line over a number or group of numbers indicates that the
number(s) are repeated indefinitely. This number or group of
numbers is called a period and they form a periodic decimal.
If we kept converting fractions and integers into decimal form, we would see that
all integers and fractions are rational numbers that can be expressed as periodic
decimals.
Remarks
1. All fractions, improper fractions and mixed fractions are rational numbers:
1
2
,
8
3
,
–2
3
7
.
Fig. 1.7 Venn diagram of ,
and
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2
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1.31
MTH-4109-1 Sets, Relations and Functions
2. All natural numbers and integers are rational numbers because they can all
be expressed as fractions:
8 =
8
1
=
16
2
= ... , –9 =
9
1
=–
18
2
= ... , 0 =
0
1
=
0
2
= ... .
3. All periodic decimals are rational numbers:
2
3
= 0.666... = 0.6 ,
1
2
= 0.5 = 0.500 0... = 0.50 .
Irrational Numbers: '
We saw earlier that all rational numbers can be expressed as periodic decimals,
but there are also numbers that are expressed as nonperiodic decimals.
Using a calculator, find the value of the following expressions.
?
2
= .......................... ;
3
=............................ ; π = ................................. .
? What can you say about the decimal expansion of these numbers?
...........................................................................................................................
In fact, these numbers do not have a period because
2
= 1.414 213 5...,
3
= 1.732 050 8... and π = 3.141 59... . These are called nonperiodic decimals.
You can also form numbers with nonperiodic decimals, such as
3.202 002 000 200 002... and 8.018 249 517 426... .
The set
' is the set of numbers that can be expressed as
nonperiodic decimals.
Since an irrational number cannot be expressed as a fraction, a number is either
rational or irrational but it cannot be both.
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1.32
MTH-4109-1 Sets, Relations and Functions
The following is a Venn diagram of sets , , and '.
Fig. 1.8 Venn diagram of
, , and '
Using a calculator, find the value of the following expressions.
?
–5
= ............................................. ;
–5
= ............................................... .
...........................................................................................................................
You probably found that
–5
= –2.236 067... . The five under the square root
sign is positive and it is the expression
–5
which is negative. In the second case,
the calculator shows the symbol E, indicating that it cannot display the result of
this operation. The square root of a negative number does not belong to
', but
rather to the set of complex numbers
, which cannot be processed by a pocket
calculator. We will not be studying this set of numbers at this level.
Real Numbers:
If we combine the set of rational numbers and the set of irrational numbers, we
obtain the set of real numbers.
The set
is the set of numbers that are rational or irrational.
= {x|x or x '}
'
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1.33
MTH-4109-1 Sets, Relations and Functions
N.B. The set of irrational numbers is written ' and corresponds to the negation
of
within the universe of real numbers .
We can finally draw the following Venn diagram of the sets
, , , ' and .
Q '
Fig. 1.9 Venn diagram of
Notes
1. Any number that belongs to also belongs to , and .
2. Any number that belongs to also belongs to and .
3. Any number that belongs to also belongs to .
4. Any number that belongs to ' also belongs to .
? a) Among the sets , , , ' and , to which set(s) does the number
4
3
belong? ......................................................
? b) In the Venn diagram below, correctly situate the following real num-
bers:
0,
3
4
,–8,π,– 9,
5
8
,2,4,–
10
2
.
1
2
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1.34
MTH-4109-1 Sets, Relations and Functions
Q '
Fig. 1.10 Venn diagram of the real numbers
In a),
4
3
is an element of and only. In b), 0 ,
3
4
, –8 , π ',
–9
= –3 ,
5
8
,
2
',
4
= 2 and
10
2
= –5 . All of this is shown
in the following Venn diagram.
Q
'
•0
••8
–9
10
2
3
4
5
8
2
4
Fig. 1.11 Venn diagram of certain real numbers
Exercise 1.6
1. Place the following real numbers in the Venn diagram below:
3,25,–
3
4
, 8, 1.372 451..., 5.7, 0.6
.
Q
'
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1.35
MTH-4109-1 Sets, Relations and Functions
2. Which of the following numbers belongs to '? ..............................................
A)
25
2
B)
–36
C)
12
D) 2.731 731 E)
–3
Did you know that...
... in France and elsewhere, the 17th century was the
golden age of mathematics? In that century, illustrious
scientists such as Fermat, Descartes, Desargues, Pascal
vancement of mathematics.
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1.36
MTH-4109-1 Sets, Relations and Functions
1.2 PRACTICE EXERCISES
1. Describe the sets below by listing their elements.
a) A = {x
|x is odd and x < –12 } A = ...............................................
b) B = {x
|–3 < x < 1} B = ...............................................
c) C = {x |x is a cube and x > 100} C = ...............................................
d) A = ...............................................
U = ..............................................
e) A = ...............................................
B = ...............................................
U = ..............................................
2. Describe the sets below by using set-builder notation.
a) A = ...............................................
U = ..............................................
?
•1
•35
•5
•7
•0
•4
A
U
•1
•35
•5
•7
•0
•4
A
U
B
•1
•5
•4
A
U
•10
•2
•20
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1.37
MTH-4109-1 Sets, Relations and Functions
b) A = ...............................................
U = ..............................................
c) C = {…, –12, –6, 0, 6, 12, ...} C = ...............................................
d) D = {–15, –14, –13} D =...............................................
e) E = {1, 3, 13, 39} E = ...............................................
3. Describe the sets below using a Venn diagram.
a) F = {0, 3, 5, 7, 9, 11} b) G = {–12, –4, 6, 9}
c) H = {x |–2 < x < 5} d) J = {x |x < 5}
e) K = {x |x < 6 and x is a multiple of 7}
•1
•5
•7
•0
•4
A
U
•10
•2
•6
•8
•3
•9
1
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3
1.38
MTH-4109-1 Sets, Relations and Functions
4. Describe sets U, A and B by listing their elements and by using set-builder
notation.
•1
•5
•7
•0
•4
A
U
B
•12
•2
•6
•8
•3
•9
•10
•11
List of elements Set-builder notation
U = ..................................... U = ..............................................................
A = ..................................... A = ..............................................................
B = ..................................... B = ..............................................................
5. Place the following real numbers in the Venn diagram below:
1.4,–
12
5
,8,–7,
8
2
, 2.16, 0.010 01...
.
Q'
6. Determine to which set(s) the following numbers belong by checking off the
appropriate boxes.
'
5/2
π
3
–7
12
3.14
25
0
1
2
3
1.39
MTH-4109-1 Sets, Relations and Functions
1.3 REVIEW ACTIVITY
1. The elements of sets U, A and B are listed below. Draw a Venn diagram of
these sets and describe them by means of set-builder notation.
U = {–1, 0, 1, 2, 3, 4, 5, 6, 7}; A = {–1, 0, 1, 2, 3}; B = {2, 3, 5, 7}.
Venn diagram Set-builder notation
U = .................................................
A = .................................................
B = .................................................
2. Describe the sets below by listing their elements and then draw a Venn
diagram of sets
, A and B.
a) A = {x
|x > 3 and x is a divisor of 6} =..................................................
b) B = {x
|1 < x < 7} =................................................................................
c) C = {x |x and x is less than 0} = ...........................................................
d) D = {x |x < 20 and x is a prime number} = ..........................................
e) Venn diagram showing the sets
, A and B.
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1.40
MTH-4109-1 Sets, Relations and Functions
1.4 THE MATH WHIZ PAGE
The Nosy Poll
The Nosy Polling Company recently conducted a survey to find out
which soft drinks Quebecers prefer. A total of 1 024 people were
interviewed across the province.
They were asked the following question: "Which soft drink did you
drink last week?" Respondents were given the following choice of
answers: PETSI, KOLA, another brand or none at all.
The following are the results:
428 people drank PETSI;
475 people drank KOLA;
285 people drank another brand of soft drink;
100 people drank PETSI and another brand;
88 people drank KOLA and another brand;
181 people drank PETSI and KOLA;
53 people drank PETSI, KOLA and another brand.
Can you determine how many people said they didn't drink any soft
drink the previous week?
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1.41
MTH-4109-1 Sets, Relations and Functions
1. Let sets P, K and O represent PETSI drinkers, KOLA drinkers and
the drinkers of other brands respectively; S is the set of all those
who took part in the survey.
2. Complete the following Venn diagram. Remember that the num-
bers shown in the diagram do not represent elements, but rather
the number of people who drank a particular soft drink.
3. Since 53 people drank PETSI, KOLA and some other brand of soft
drink, we can write 53 in the intersection of the three sets, as
shown in the diagram below.
4. Since 181 people drank PETSI and KOLA, there are just
128 (181 – 53) who drank only PETSI and KOLA. This number is
shown in the appropriate part of the Venn diagram.
S
P
K
A
128
53
Fig. 1.12 Venn diagram of the survey results
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1.42
MTH-4109-1 Sets, Relations and Functions
Now you should be able to do the rest of the problem on your own and
come up with the right answer!
................................................................................................................
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