MTH-4109-1 Sets, Relations Functions · PDF fileCartesian Product.....5.1 6. Relations ......

MTH-4109-1

Sets, 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 allcountries. Any reproduction, by mechanical or electronic means, including microre-production, is forbidden without the written permission of a duly authorizedrepresentative of the Société de formation distance des commissions scolaires duQuébec (SOFAD).

Legal Deposit — 2007

Bibliothèque et Archives nationales du Québec

Bibliothèque et Archives Canada

ISBN 978-2-89493-287-2

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MTH-4109-1 Sets, Relations and Functions

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TABLE OF CONTENTS

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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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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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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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-4108-1 Quadratic Functions

MTH-4107-1 Straight Lines II

MTH-4106-1 Factoring and Algebraic Functions

MTH-4105-1 Exponents and Radicals

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

TradesDVS

MTH-5112-1 Logic

25 hours = 1 credit

50 hours = 2 credits

MTH-4104-2 Statistics II

You ar e here

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Hi! My name is Monica and I have beenasked to tell you about this math module.What’s your name?

I’m Andy.

Whether you areregistered at anadult educationcenter or pur-suing distanceeducation, ...

You’ll see that with this method, math isa real breeze!

... you have probably taken aplacement test which tells youexactly which module youshould start with.

My results on the testindicate that I should beginwith this module.

Now, the module you have in yourhands is divided into threesections. The first section is...

... the entry activity, whichcontains the test on theprerequisites.

By carefully correcting this test using thecorresponding answer key, and record-ing your results on the analysis sheet ...

HOW TO USE THIS GUIDE

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?

The memo pad signals a brief reminder ofconcepts which you have already studied.

The calculator symbol reminds you thatyou will need to use your calculator.

The sheaf of wheat indicates a review designed toreinforce what you have just learned. A row ofsheaves near the end of the module indicates thefinal review, which helps you to interrelate all thelearning activities in the module.

The starting lineshows where thelearning activitiesbegin.

The little white question mark indicates the questionsfor which answers are given in the text.?

... you can tell if you’re well enoughprepared to do all the activities in themodule.

The boldface question markindicates practice exerciseswhich allow you to try out whatyou have just learned.

And if I’m not, if I need a littlereview before moving on, whathappens then?

In that case, before you start theactivities in the module, the resultsanalysis chart refers you to a reviewactivity near the end of the module.

In this way, I can be sure Ihave all the prerequisitesfor starting.

Exactly! The second sectioncontains the learning activities. It’sthe main part of the module.

Look closely at the box tothe right. It explains thesymbols used to identify thevarious activities.

The target precedes theobjective to be met.

Good!

?

START

Lastly, the finish line indicatesthat it is time to go on to the self-evaluationtest to verify how well you have understoodthe learning activities.

FINISH

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A “Did you know that...”?

Later ...

For example, words in bold-face italics appear in theglossary at the end of themodule...

Great!

... statements in boxes are importantpoints 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 differentparts of the module.

Yes, for example, short tidbitson the history of mathematicsand fun puzzles. They are in-teresting and relieve tension atthe same time.

No, it’s not part of the learn-ing activity. It’s just there togive you a breather.

There are also many fun thingsin this module. For example,when you see the drawing of asage, it introduces a “Did youknow 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 hasbeen arranged to makelearning easier.

There is also a self-evaluationtest and answer key. They tellyou if you’re ready for the finalevaluation.

Thanks, Monica, you’ve been abig help.

I’m glad! Now,I’ve got to run.

See you!This is great! I never thought that I wouldlike mathematics as much as this!

They are so stimulating thateven if you don’t have to dothem, you’ll still want to.

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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

your own learning.

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INTERMEDIATE AND TERMINAL OBJECTIVES OFTHE 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 1 5%

12 and 13 1 5%

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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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, n1 < x < n2, x ≤ n, x ≥ n, n1 ≤ x ≤ n2,

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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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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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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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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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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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 = ax2 + bx+ c or

y = a(x – h)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(x – h)2 + k, a ≠ 0) and

the general form ( f (x) = ax2 + bx + c, a ≠ 0) of the equation of a quadratic

function.

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21. Determining the equation of a function given a written statement

Given a written statement providing relevant information about afunctional 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 followingfunctions:• 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 ofGraph

Rule orvalues 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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(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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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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DIAGNOSTIC TEST ON THE PREREQUISTES

Instructions

1. Answer as many questions as you can.

2. You may use a calculator.

3. Write your answers on the test paper.

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

your answers using the answer key which follows the diagnostic

test.

6. To be considered correct, your answers must be identical to

those in the key. In addition, the various steps in your solution

should be equivalent to those shown in the answer key.

7. Transcribe your results onto the chart which follows the answer

key. This chart gives an analysis of the diagnostic test results.

8. Do the review activities that apply to each of your incorrect

answers.

9. If all your answers are correct, you may begin working on

this module.

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1

y

xE

D

F

CB

0 1

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

A B C D E 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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• •

•

•

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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ANSWER KEY FOR THE DIAGNOSTIC TESTON 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

A B C D E 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 = y2 – y1x2 – x1

= 3 – 00 – 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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b) Equation of straight line CD

Given points C(–2, –1) and D(1, 3), calculate the slope.

m = y2 – y1x2 – x1

= 3 – (–1)1 – (–2)

= 3 + 11 + 2 = 4

3

Hence, 43 = y – 3

x – 13(y – 3) = 4(x – 1)

3y – 9 = 4x – 4

3y = 4x – 4 + 9

3y = 4x + 5

y = 43 x + 5

3

The equation of straight line CD is therefore y = 43 x + 5

3 .

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ANALYSIS OF THE DIAGNOSTICTEST RESULTS

QuestionsAnswers Review Before Going to

Correct Incorrect Section Page Unit(s)

1. a) 15.1 15.4 Unit 1b) 15.1 15.4 Unit 1c) 15.1 15.4 Unit 1d) 15.1 15.4 Unit 1e) 15.1 15.4 Unit 1

2. a) 15.2 15.11 Unit 4b) 15.2 15.11 Unit 4c) 15.2 15.11 Unit 4

3. a) 15.3 15.14 Unit 5b) 15.3 15.14 Unit 5c) 15.3 15.14 Unit 5d) 15.3 15.14 Unit 5e) 15.3 15.14 Unit 5f) 15.3 15.14 Unit 5

4. a) 15.3 15.14 Unit 6b) 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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INFORMATION FOR DISTANCEEDUCATION 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

tutor will correct your work and help you with your studies. Do not hesitate to

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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The following guidelines concerning theory, examples, exercises and assign-

ments are designed to help you succeed in this mathematics course.

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

difficulty hinders your progress, contact your tutor before handing in your

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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Exercises

The exercises in each unit are generally modeled on the examples provided. Here

are a few suggestions to help you complete these exercises.

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,

even if your answers are correct.

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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1. Do a rough draft first, and then, if necessary, revise your solutions before

writing out a clean copy of your answer.

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

answered by your tutor. He or she will give you advice and guide you in your

studies, if necessary.

In this course

Homework Assignment 1 is based on units 1 to 7.



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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UNIT 1

SETS OF NUMBERS ANDDESCRIPTIONS 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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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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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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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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: 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, ..., – 12 , ..., 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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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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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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? 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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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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g) G = {..., –3, –2, –1, 0, 1, 2, 3, ...}

.......................................................................................................................

h) H = {0, 2, 4, ..., 94, 96, 98}

.......................................................................................................................

3. Our typographer had a bad day and made a few mistakes listing the elements

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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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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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) 12 ∉ 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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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) 0–5 ...... 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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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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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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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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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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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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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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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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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

•–2A

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•0

•1•2

•4

•5•6

•7•3

A B

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.

A B

Fig. 1.3 Venn diagram

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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

•4A 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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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}

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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

...........................................................


...........................................................

c) • Listing of elements

...........................................................


...........................................................

•7

A•3

•5

•9•11 •13

B

•14

•0•7

•35

•28•21

•1

•4

•0C

•9•36

•16

•25

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d) • Listing of elements

...........................................................


...........................................................

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

•0D

•–1•–3•–2

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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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b) Represent all of these sets in the Venn diagram below.

UA 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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• 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.

= { ab |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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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.

13 = 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; 14 = 0.25 = 0.250 00... and is written 0.250;

– 2 17 = –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:

12 ∈ , – 8

3 ∈ , – 2 37 ∈ .

Fig. 1.7 Venn diagram of , and

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2. All natural numbers and integers are rational numbers because they can all

be expressed as fractions:

8 = 81 = 16

2 = ... ∈ , –9 = – 91 = – 18

2 = ... ∈ , 0 = 01 = 0

2 = ... ∈ .

3. All periodic decimals are rational numbers:

23 = 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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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 = ............................................... .

? What are your findings? ...................................................................................

...........................................................................................................................

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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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 – 43

belong? ......................................................

? b) In the Venn diagram below, correctly situate the following real num-

bers: 0, – 34 , – 8, π, – 9 , 5

8 , 2, 4 , – 102 .

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Q '

Fig. 1.10 Venn diagram of the real numbers

In a), – 43 is an element of and only. In b), 0 ∈ , – 3

4 ∈ , –8 ∈ , π ∈ ',

– 9 = –3 ∈ , 58 ∈ , 2 ∈ ', 4 = 2 ∈ and – 10

2 = –5 ∈ . All of this is shown

in the following Venn diagram.

Q '

•0

•

•

• • –8 •

•

•

– 9

– 102

– 34

58

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 , – 34 , – 8, 1.372 451..., – 5.7, 0.6.

Q '

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2. Which of the following numbers belongs to '? ..............................................

A) 252 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

and Bernouilli made remarkable contributions to the ad-

vancement of mathematics.

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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

UB

•1 •5•4

A

U

•10•2

•20

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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 •4A

U

•10

•2 •6

•8•3 •9

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4. Describe sets U, A and B by listing their elements and by using set-buildernotation.

•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, 82 , – 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

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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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1.4 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?

The following procedure will help you answer this question.

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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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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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