IB Diploma Program Mathematics Course Companion Higher Level Option: Sets, Relations and Groups [1 ed.] 0198304862, 9780198304869

Citation preview

O X

F O

R

D

I B

D

I P L O

M

A

M AT H E M AT I C S

SETS,

AND

P

R

O

G R

A

M

M

E

HI GH E R

LE V E L

R E L AT I O N S

GR O U P S

Josip Harcet

Lorraine Heinrichs

Palmira Mariz Seiler

Marlene Torres-Skoumal

:

3

Great

Clarendon

Oxford

It

furthers

and

the

Oxford

New

Auckland

New

©

the

is

Oxford

The

Database

First

All

rights

in

without

or

in

as

right

the

must

and

you

British

Data

of

Oxford.

research,

scholarship,

in

Hong

Kong

City

Karachi

Nairobi

No

Turkey

of

Oxford

France

Poland

Ukraine

Greece

Portugal

Vietnam

University

Press

2015

have

University

part

of

been

Press

this

or

permission

permitted

asserted

(maker)

of

by

the

Press,

in

writing

law,

impose

above

at

or

book

Cataloguing

in

under

be

address

same

of

in

may

in

any

Oxford

terms

Enquiries

should

the

this

this

publication

transmitted,

organization.

circulate

Library

Korea

countries

authors

system,

rights

scope

not

mark

other

Republic

South

2015

prior

must

Czech

be

reproduced,

form

or

agreed

with

concerning

sent

to

by

University

the

any

means,

Press,

the

appropriate

reproduction

Rights

Department,

above

any

other

condition

Publication

on

binding

any

or

cover

acquirer

Data

978-0-19-830486-9

9

8

Printed

Paper

The

in

available

ISBN

10

University

Toronto

Thailand

Press

Oxford

University

You

the

Mexico

Japan

trade

the

retrieval

expressly

Oxford

of

reserved.

reprographics

outside

of

excellence

Salaam

Chile

Italy

certain

rights

the

es

Brazil

University

a

of

worldwide

Taipei

registered

published

stored

objective

in

and

moral

6DP

Melbourne

Switzerland

a

UK

OX2

department

Dar

Hungary

Singapore

Oxford

Town

Madrid

Austria

Guatemala

a

publishing

Shanghai

oflces

is

York

Lumpur

Argentina

in

by

Cape

Delhi

With

Oxford

Press

University’s

education

Kuala

Street,

University

7

in

6

5

4

Great

used

in

3

2

1

Britain

the

by

Bell

production

manufacturing

process

and

of

Bain

this

Ltd,

book

conforms

to

is

the

Glasgow

a

natural,

recyclable

environmental

product

regulations

made

of

the

from

wood

country

of

grown

in

sustainable

forests.

origin.

Acknowledgements

The

p3:

publisher

Trajano

would

like

Paiva/Alamy;

Collection/Archives

to

thank

p11:

the

Diego

following

for

permission

Cervo/Shutterstock;

Charmet/Bridgeman

Images; p88:

to

reproduce

p12l-r:

Eric

photographs:

Pradel;

snig/Shutterstock;

p14t:

p126:

Eric

Science

Pradel;

Photo

p14b:

Library.

OUP;

p47:

Private

Course

The

IB

Diploma

resource

study

gain

an

Programme

materials

throughout

of

Companion

in

their

a

designed

two-year

par ticular

understanding

Course

to

subject.

of

Companions

suppor t

Diploma

what

are

will

help

expected

mindedness,

Programme

students

Programme

They

is

deni tion

course

students

from

the

extended

of

an

IB

Diploma

Programme

subject

content

in

a

way

that

illustrates

aims

of

the

IB.

They

reect

the

book

the

each

of

the

and

subject

IB

providing

The

books

and

encourage

of

IB

The

a

in

wide

to

this

and

terms

range

Inter national

create

philosophy

a

deep

end

The

a

of

of

of

philosophy

a

IB

all

and

can

help

to

and

more

wider

of

encouraged

lear ners

minded

strive

with

aims

the

approach;

to

are

In

addition,

the

They

the

and

a

better

to

throughout

local

They

and

in-depth

each

are

required

a

and

variety

of

fur ther

book

and

suggestions

for

how

provided.

the

on

Course

the

Companions

specic

course

provide

assessment

honesty

protocol.

being

programmes

people

world

who

They

advice

and

requirements

are

distinctive

prescriptive.

of

inter national

education

and

rigorous

assessment.

through These

programmes

world

to

lear ners

gover nments

develop

challenging

become

who

dierences,

is

to

They

develop

recognizing

guardianship

of

take

encourage

active,

students

across

compassionate,

understand

can

also

be

that

other

and

the

people,

lifelong

with

their

right.

responsibility

consequences

their

that

and

more

for

their

accompany

own

actions

and

the

them.

the

peaceful

They

understand

and

appreciate

their

world. cultures

and

their

natural

curiosity .

perspectives,

to

conduct

inquir y

personal

histories,

and

are

open

and

this

in

lear ning.

love

of

their

and

traditions

of

to

other

They and

and

communities.

evaluating

a

They

range

of

are

accustomed

points

of

view ,

to

and

They

lear ning

will

willing

to

grow

from

the

experience.

be

They

show

empathy ,

compassion,

and

respect

lives.

explore

global

values,

and

concepts,

ideas,

and

the

signicance.

In

so

doing,

knowledge

and

develop

needs

and

feelings

of

others.

They

have

a

issues commitment

to

ser vice,

and

act

to

make

a

they positive

acquire

from

additional

without

personal have

for

academic

towards

that

other

are

Prole

independence

lear ning

Knowledgable

conclusions

authoritative

Caring

sustained

with

IB

be:

necessar y

show

enjoy

the

on

are

actively

of

and

seeking

research

conjunction

respect.

who,

shared

develop

skills

draw

in

research

individuals

acquire

in

students

and

the

Inquirers

to

given

extend

guidance

develop

young

schools,

people

and

to

used

Suggestions

to

own

IB

ser vice

issues

viewing

peaceful

and

prog rammes

create

be

indeed,

Open-minded

planet,

activity ,

and

inter national

caring

organizations

humanity

Diploma

statement

works

IB

creativity ,

IB

knowledge,

thinking.

whole-course

resources,

the

of

understanding

to

critical

learner

inter nationally

common

for

Baccalaureate

better

the

IB

aim

connections

understanding

inter national

The

IB

knowledgable

intercultural

To

the

mission

inquiring,

help

making

opportunities

mirror

curriculum

use

by

and

and

theor y

pur pose

reading

of

essay ,

and

resources.

approach

prole

while

and

and

lear ner

requirements,

(CAS).

materials

presenting

IB

the Each

study

the

core

dierence

to

the

lives

of

others

and

to

the

understanding environment.

across

a

broad

and

balanced

range

of

disciplines.

Risk-takers Thinkers

They

exercise

initiative

in

applying

uncer tainty skills

critically

and

creatively

to

recognize

and

problems,

and

make

reasoned,

approach

unfamiliar

situations

and

with

courage

and

forethought,

and

have

approach

the complex

They

thinking

independence

of

spirit

to

explore

new

roles,

ideas,

ethical

and

strategies.

They

are

brave

and

ar ticulate

in

decisions.

defending Communicators

They

understand

and

express

Balanced

and

information

condently

and

creatively

in

their

They

one

language

and

in

a

variety

of

modes

collaboration

Principled

strong

They

sense

dignity

of

of

the

They

with

work

eectively

and

willingly

act

with

individual,

well-being

Reective

integrity

justice,

and

and

groups,

honesty ,

respect

and

and

the

impor tance

emotional

of

balance

for

with

the

communities.

for

themselves

and

to

achieve

others.

in

others.

fair ness,

physical,

of

personal communication.

understand

more

intellectual, than

belief s.

ideas

a

own

They

learning

understand

support

and

their

their

give

thoughtful

experience.

strengths

learning

and

consideration

They

and

are

able

limitations

personal

to

in

to

their

assess

order

and

to

development.

iii

A

It

note

is

of

vital

all,

that

academic

impor tance

appropriately

when

on

credit

the

information

owners

of

ideas

to

acknowledge

owners

is

used

of

in

(intellectual

honesty

and

What constitutes misconduct?

information

your

work.

proper ty)

Misconduct

is

result

or

in,

you

work,

it

rights.

must

original

To

be

ideas

have

based

with

an

on

the

authentic

your

work

piece

individual

of

others

Therefore,

all

oral,

completed

for

in,

an

or

may

unfair

in

one

or

more

assessment

component.

Misconduct

includes

Plagiarism

is

plagiarism

and

collusion.

dened

as

the

representation

of

the

fully

assignments,

assessment

results

gaining

and

must

or

work

of

another

person

as

your

own.

written

The or

that

student

of

ideas acknowledged.

any

have advantage

proper ty

behaviour

After

use

following

are

some

of

the

ways

to

avoid

your

plagiarism: own

language

used

or

and

referred

quotation

or

expression.

to,

whether

paraphrase,

appropriately

Where

in

the

such

sources

form

sources

of

are

direct

must

●

be

do

work

I

of

and

suppor t

acknowledged.

ideas

one’s

of

another

arguments

person

must

used

to

be

acknowledged.

●

How

Words

acknowledge

the

Passages

that

are

enclosed

within

quoted

verbatim

quotation

marks

must

be

and

others? acknowledged.

The

way

that

you

acknowledge

that

you

have ●

used

the

ideas

of

other

people

is

through

the

CD-ROMs,

Inter net, of

footnotes

and

endnotes

(placed

(placed

at

at

the

the

bottom

end

of

a

of

a

page)

document)

or

provided

and

are

when

you

quote

or

paraphrase

treated

in

any

the

document,

or

information

provided

do

to

The

sources

of

closely

summarize

in

another

document.

is

par t

of

provide

a

a

footnote

for

do

are

the

par t

of

‘body

not

of

need

knowledge’.

to

assumed

Bi bliographies

That

should

be

resources

that

you

Words

footnoted

as

should

include

magazines,

resources,

include

used

all

in

a

formal

your

resources,

newspaper

CDs

list

that

accepted

full

of

A

you

and

forms

should

of

information

work.

your

work

can

bibliography

essay .

is

to

works

use

how

nd

computer

and

programs,

similar

if

they

data,

material

are

of

ar t,

ar ts,

whether

or

not

music,

visual

ar ts,

a

of

use

must

acknowledged.

must

graphs,

be

your

own

work.

lm,

and

dance,

where

the

of

par t

a

work

takes

place,

be

of

one

of

the

a

same

compulsor y

Y ou

as

suppor ting

misconduct

by

student.

This

includes:

allowing

several

must

or

the

your

work

to

be

copied

or

submitted

assessment

by

another

student

‘Formal’

provide

viewer

information.

in

dened

Inter net-

ar t.

the

reader

is

The

extended

duplicating

work

components

Other

that

forms

gives

results

taking

room,

of

you

of

an

diploma

unfair

requirements.

include

during

or

Examples

material

record.

assessment

advantage

student.

misconduct

CAS

dierent

misconduct

another

a

for

and/or

unauthorized

falsifying

iv

maps,

including

ar ticles,

presentation.

as

photographs,

of

●

means

all

creative

for based

must

and

knowledge.

●

books,

media

books

they

another

listing

as

is,

Collusion

the

electronic

way

information

theatre denitions

other

same

Y ou

●

that

the

the

acknowledged

need

on

from

audio-visual,

not

sites

to

illustrations,

another

web

jour nals.

●

be

messages,

bibliographies. be

Footnotes

email

use

an

into

an

any

action

aects

the

include,

examination

examination,

and

About

The

new

Option:

Each

with

the

syllabus

Sets

is

chapter

the

for

book

Mathematics

thoroughly

is

divided

following

Higher

covered

into

in

this

lesson-size

Level

book.

sections

Questions

strengthen

through

given

you

analytical

appropriate

in

the

style

Note: Companion

emphasis

is

improved

concepts

the

will

prociency

thinking.

guide

in

She

you

years

been

this

moderator

workshop

member

Palmira

for

of

the

Mariz

community

in

Inter national

as

workshop

HL

Bonn

IB

DP

the

over

2001

are

calculator.

is

a

growing,

contextual,

enables

ever

technology

students

to

become

US

spelling

review

has

been

used,

with

IB

style

for

terms.

and

well

as

denotes

for

IB

examination

may

be

used.

a

she

was

review

been

since

since

at

deputy

Colombiano

in

the

IB

Vienna

also

as

in

a

examiner

teaches

Bogota,

enjoyed

including

and

a

for

over

various

deputy

chief

moderator

for um

member

30

of

for

has

taught

years.

roles

During

with

examiner

Inter nal

moderator,

several

IB

the

for

this

time,

IB,

HL,

Assessment,

workshop

curriculum

leader,

review

teams.

moderator

chief

she

has

Josip

has

and

she

calculator

a

joined

the

Torres-Skoumal

mathematics

senior

and

also

then

groups

Currently

2002.

team.

She

teacher

working

School.

teaching

years.

and

for

senior

Assessment

IB;

25

Marlene

mathematics

been

Assessment

and

lear ners.

The

as

Inter national

also

has

as

School

Anglo

examples

critical

coordinator

has

Seiler

leader

to

display

all

teaching

and

Inter nal

of

and

GDC

been

at

mathematics.

Colegio

a

years

Inter nal

curriculum

solutions

graphics

authors

has

she

application

suitable

curriculum

for

condence

mathematical

solving

be

of

development

life

30

HL

leader

mathematics

worked

the

time

coverage

Companion

the

for

build

diculty ,

through

assessment.

of

where

Heinrichs

16

has

During

would

those

mathematics

past

real

Course

About

the

the

problem

that

and

Lorraine

on

their

full

inter nal

placed

The

questions

practice

new

with

understanding

and

The

lifelong

mathematical curriculum

and

a

approach

adaptable,

topics

and

in

Advice

Extension

latest

the

of

education

entity .

integrated

the

skills

increase

History

know?

changing

Course

to

understanding.

Mathematics

The

designed

features: Where

Did

are

the

Harcet

IB

has

programme

curriculum

review

examiner

for

examiner

and

HL

as

been

well

since

senior

a

1992.

member,

Fur ther

as

involved

with

He

has

deputy

Mathematics,

examiner

workshop

for

leader

and

teaching

ser ved

as

a

chief

assistant

IA

Mathematics

since

1998.

for

at

Colombia.

v

Contents

Chapter

1

The

The

Introduction

1.1

Set

development

language

denitions

and

Well-dened

1.2

Par titions

1.3

Venn

1.4

The

1.5

Relations

Set

and

2

and

Inverse

classes

of

two

sets

21

25

and

27

par titions

32

of

the

of

concept

the

function

of

function

concept

relations

of

of

of

functions

59

61

66

70

operations

72

binar y

The

identity

The

inverse

The

cancellation

operations

element

of

an

76

e

78

element

79

laws

81

exercise

Chapter

3

83

The

in

‘Universal

Theory

of

Everything’

Mathematics

Group

Introduction

86

Theor y

88

Groups

89

Innite

Finite

groups

of

Proper ties

groups

exercise

theorems

left

Subgroups

Review

modulo

n

98

groups

and

and

94

integers

Symmetr y

Cyclic

90

groups

Groups

Right

47

50

functions

of

46

48

functions

functions

Proper ties

vi

14

functions

Proper ties

Binar y

5

12

proper ties

Congr uence

as

Identity

3.3

dierence

relations

Composition

3.2

set

16

Evolution

Equality

3.1

4

and

42

Functions

Review

sets

diagrams

set

product

Extension

Introduction

2.3

3

exercise

Chapter

2.2

sets

2

23

Modular

2.1

Theory

proper ties

Car tesian

Equivalence

Review

equal

Venn

Equivalence

1.6

Set

operations

sets,

diagrams

of

of

100

of

groups

cancellation

laws

and

for

subgroups

groups

105

105

108

114

119

Chapter

4

The

Introduction

4.1

classication

Group

Permutation

str uctures

Proper ties

of

and

Cosets

4.3

Homomor phisms

4.4

of

Isomor phisms

Review

exercise

a

form

form

Lagrange’s

ker nel

124

126

126

cycle

cycle

4.2

The

groups

groups

Permutations

and

of

theorem

130

132

135

139

homomor phism

142

144

153

Answers

156

Index

165

vii

The

development

1 of

CHAPTER

F inite

8.1

and

innite

Morgan’ s

and

Before

Given

subsets;

difference,

laws:

pairs:

relations,

1

set

sets;

Operations

symmetric

distributive,

on

α,

the

Car tesian

equivalence

you

that

difference;

associative

and

β

product

classes

and

of

two

of

α(

−

union,

Venn

intersection,

diagrams;

commutative

sets;

laws

for

union

z

equivalence

start

are

the

roots

of

the

1

a

Given

that

α,

β

are

the

roots

of

the

equation

2

−

α)

relations:

par titions.

2

equation

sets:

intersection.

Ordered

8.2

Theory

OBJECTIVES:

complement,

De

Set

4z

+

β(

+

13

−

=

β ),

0,

nd

the

without

value

−

z

4z

+

1

=

0,

solving

2

the

quadratic

equation.

1

⎛

nd

the

value

of

⎜

α

⎟

α

⎝

Using

Viete’s

dierence

of

formulas

for

sum

2

⎞

1

⎛

+

β ⎜

⎠

⎞

⎟

β

⎝

⎠

and

roots: 2

b

α

β

+

=

4,

α β

=

3

α

If

and

show

β

that

are

the

the

roots

roots

of

of

the

α

α (

−

α)

+

β(

8z

+

7z

+

8

=

0

=

α

−

α

+

β

either

2

=

α

+

β

−

(α

+

β

=

α

+

β

−

((α

+

β )

=

4

)

2

2

The

(6

without α

β

−

−

26)

development

=

of

+

4

=

0,

β

2

2

−

3x

equation

and

are β

2

+

2

β )

−

2x

−

2αβ )

4

Set

Theory

of

the

two

given

equations.

solving

The

In

language

of

this

chapter

we

Georg

Cantor,

9th

known

of

a

for

set

objects

as

of

his

“...

our

between

will

be

creation

the

of

taking

and

looking

centur y

intuition

sets,

sets

to

or

the

into

he

a

He

basic

elements

mathematician

language

thought”.

this

the

German

together

do

at

of

sets,

whole

went

of

on

associated

to

is

theor y .

best

the

notion

well-dened

study

each

set

who

explained

distinct

with

of

the

set

relation

a

cardinal

A

number

which

would

but

innite

help

him

compare

sizes,

not

only

of

nite

cardinal

is

also

ones.

Stated

simply ,

by

comparing

dierent

one

Cantor

discovered

that

there

are

dierent

sizes

of

innite

elements,

which

is

numbers

is

size

of

the

smaller

than

continuous.

are

all

(cardinality).

said

He

set

the

The

to

be

called

of

Natural

innite

Natural

size

size

of

of

the

numbers,

countable,

the

numbers,

up

set

real

of

Integers

innite

the

made

and

countable

of

and

have

or

an

of

something.

discrete

numbers,

Rational

the

innite

quantity

innity . amount

The

which

innite denotes

sequences

number

sets

same

size

whereas

sets 0

the

innity

associated

with

the

uncountable

real

numbers

was

. 1

He

fur ther

made

Hypothesis.

is

In

a

his

conjecture

conjecture

and

between

.

0

Hypothesis

problems

worked

work

20th

at

the

the

rst

tur n

extensively

centur y

became

never

says

known

that

proved

as

there

this,

the

is

and

Continuum

no

the

set

whose

size

Continuum

1

was

changed

Cantor

that

Cantor

the

and

of

on

on

the

this

focus

of

opened

the

famous

20th

David

centur y .

conjecture

to

list

Gödel

and

Kur t

between

mathematics

doors

Hilber t

many

in

the

other

930

and

second

of

unsolved

Paul

966.

half

of

Cohen

Their

the

theories.

Chapter

1

3

The

Hilber t

German

it

adver tised

Hotel:

ever ything

anyone

wanting

an

does

At

innite

the

this

was

twice

there

hotel,

1.1

with

Much

since

of

to

its

that

his

it

number

the

is

rst

the

and

is

basic

and

and

of

this

always

help

move

in

to

difcult

each

its

This

way

nd

buses

bus?

different

Y ou

his

room

warn

a

for

was

for

solving

did

to

of

room

the

opened,

busy

it

full!

one

the

way

show

How

more?

of

up

that

odd

allocating

want

chapter

of

in

you

most

the

will

of

Prior

the

to

the

you

have

Higher

Level

syllabus.

A

set

we

S

say

For

The

is

we

(S )

will

or

nite

set

whose

number

The

is

set

is

of

A

an

the

of

of

objects,

element

subjects

elements

denote

it

by

of

S.

and

We

oered

in

a

set

n(S).

In

in

S

if

x

is

one

denote

the

is

IB

these

by

x

diploma

called

some

of

this

the

books

objects

∈ S

form

a

cardinality

it

is

denoted

set.

of

the

empty

The

is

is

one

with

cardinality

elements

=

{l,

3,

exactly

set,

set

by

|S|.

5,

a

is

nite

a

then

7,

9}

number

natural

we

is

say

elements,

number.

that

nite

of

the

If

set

whereas

a

is

the

set

i.e.

has

a

nite

an

set

one

denoted

development

set

by

of

that

∅

Set

=

has

{}.

Theory

no

elements

is

innite

innite.

set B

=

{2,

4,

6,

8,

innite.

There

4

x

number

card

one

collection

example,

and

A

a

that

and

we

“In

ar t

asking

is

of

more

solving

call

this

the

...}

as

Theor y.

thesis

was

the

questions

valuable

than

problems”

David

(1862−1943)

said

Cantor’ s

“the

nest

and

known

Set

mathematics

one

of

intellectual

encountered,

mathematics

of

doctoral

titled,

activity. ”

already

Lear ning

His

is

of

work

product

was,

of

genius

the

achievements

the

aspects!

have

founder

mathematical

that

at

Cantor

the

Hilber t

hotel

Hilber t

might

time,

number

all

Georg

the

guest!

rooms

so

by

operations

language

included

for

number

in

all

is

rst

more

especially

hotel,

however

ver y

people

an

the

one

enough

occupied.

Cantor

innite

problem,

par t

also

originally

an

of

there

should

be

in

created

Innity

for

than

was

staying

Cantor

guest

would

suppose

innite

friend

it

Hotel

room

more

that

experiment

When

has

however ,

available.

when

were

were

they

thought

always

promise,

each

a

Hilber t.

there

week,

number

intriguing

is

that

people

denitions

sets

studied,

an

of

became

example,

this

Set

room

as

One

asked

situations

For

research

up

said

rooms

were

rooms.

live

David

hotel

ne,

stay.

Hilber t

the

the

was

to

Cantor

numbered

as

number

hotel

point,

problem.

Innity

mathematician

itself

Initially

and

Hotel

supreme

of

purely

human

Set

builder

notation

is

a

mathematical

notation

used

to

describe Set

sets,

whether

nite

or

innite.

The

following

builder

consists

illustrate

this:

within

=

{l,

3,

A

=

{x | x

5,

7,

9}

=

2n

−

in

set

builder

notation

of

three

curly

variable,

A

notation

examples

a

par ts

brackets:

ver tical

a

line

becomes (or

a

colon)

and

any

+

1,

n

∈ 

in

set

,

n

≤

5}

restrictions

on

the

variable.

B

=

{2,

4,

B

=

{x | x

6,

8,

...}

builder

notation

becomes

+

Y ou

=

have

jour ney

2 n,

been

so

far.

The

natural

The

integers

The

positive

The

negative

The

rational

n

∈ 

using

Here

}

a

is

number

a

list

of

of

innite

them

sets

using



numbers



in

the

your

IB

= {0, 1 ,

= {0,

mathematical

symbols

2,

± 1 ,

for

the

sets:

3}

± 2,

± 3, }

+

integers



integers



numbers



= {1 ,

=

2,

{1,

⎧

3,  }

 2,

 3,

 }

⎫

p

+

=

p,

⎨

q

∈ ,

q

≠

0

Note

⎬

that

Q

can

also

be

q ⎩

⎭ described

⎧

positive

rational

numbers



+



=

p,

⎨

q

∈ 

⎧ p

+

⎫

p

+

The

⎬

as

=

⎩

−

p

⎨ q

q

∈



⎫

⎬

⎭

q ⎩

The

real

numbers,

denoted

by

R,

are

often

⎭

represented

√2

by

a

number

line.

0

+

The

positive

real

The

complex



numbers



numbers

Well-dened

sets,

equal

sets

and

= {x | x

=

set

{a

+

∈

,

ib | a,

x

b

∈

>

0}

,

i

=

−1 }

dierence

Denition

A

set

S

is

said

determine

if

to

x

be

well-dened

belongs

to

the

if

for

any

given

x,

we

can

set.

+

For

example,

P

=

{n|n

∈



,

n