IB Diploma Program Mathematics Course Companion Higher Level Option: Discrete Mathematics [1 ed.] 0198304870, 9780198304876

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O X

F O R

D

I B

D

I P L O

M

A

M AT H E M AT I C S

P

R

O

G R

A

M AT H E M AT I C S

C O M PA N I O N

Josip Harcet

Lorraine Heinrichs

Palmira Mariz Seiler

Marlene Torres-Skoumal

M

E

HI GH E R

DI S C R E T E

C O U R S E

M

LE V E L

:

3

Great

Clarendon

Oxford

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furthers

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v

Contents

Chapter

1

Making

A

Introduction

1.1

Number

1.2

Integers,

brief

systems

prime

Diophantus

Linear

Prime

1.3

Strong

1.4

The

of

different

number

systems

2

factors

and

13

divisors

20

21

equations

26

30

induction

Theorem

of

Arithmetic

and

33

multiples

37

Modular

From

Congr uence

2.2

Modular

2.3

The

Pigeonhole

2.4

The

Chinese

2.5

Using

ari thmetic

Gauss

modulo

inverses

to

and

i ts

applications

40

41

cr yptography

42

n

and

linear

48

congr uences

53

Principle

Remainder

Theorem

or

systems

of

57

congr uences

cycles

for

powers

modulo

n

and

Fermat’s

Little

Theorem

3

Recursive

patterns

Modelling

Introduction

and

Recurrence

relations

3.2

Solution

first-degree

3.3

Modelling

of

applications

to

Financial

and

sequences

recurrence

relations

and

constant

83

recurrence

relations

90

91

interest

92

problems

homogeneous

coefficients

89

89

compound

linear

77

and

problems

first-degree

probability

Second-degree

exercise

using

amor tizations

Investments

with

problems

problems

and

Games

solving

linear

counting

with

76

78

3.1

Loans

64

72

exercise

Chapter

3

4

bases

numbers

2.1

Review

through

2

Alexandria

mathematical

common

linear

vi

and

numbers,

Diophantine

Introduction

3.4

numbers

exercise

Chapter

Review

of

jour ney

Fundamental

least

Review

sense

recurrence

relations

94

99

Chapter

4

From

Terminology

What

4.2

puzzles

Introduction

Introduction

4.1

folk

is

a

and

Weighted

Directed

Simple

of

a

graph

classification

graph

Classification

to

to

and

what

new

of

mathematics

104

graphs

its

104

elements?

108

graphs

108

graphs

109

graphs

109

graphs

Connected

110

graphs

111

Trees

Complete

Bipar tite

4.3

Different

4.4

Planar

same

115

graph

119

of

relation

life

Hamiltonian

Eulerian

119

graphs

for

planar

application

4.6

–

120

graphs

The

soccer

124

ball

126

cycles

circuits

and

129

trails

133

exercise

Chapter

5

Applications

Fur ther

Introduction

Graph

Shor test

Chinese

5.3

Travelling

The

of

exercise

141

Dijkstra’s

142

142

Problems

Dijkstra’s

146

147

Postman

149

151

algorithm

153

Problem

Neighbour

ver tex

148

Algorithm

problem

Salesman

Nearest

Deleted

methods

and

and

140

Problems

postman

Chinese

Theory

Algorithm

Limitation

5.2

Graph

Kr uskal’s

Connector

Path

Dijkstra’s

of

algorithms

algorithms:

Minimum

Review

the

trees

4.5

5.1

of

118

Complements

Review

113

representations

Spanning

Real

112

graphs

graphs

graphs

Euler

102

103

theor y

of

are

branch

Algorithm

algorithm

for

lower

for

upper

bound

bound

154

155

160

Answers

165

Index

177

vii

Making

1

CHAPTER

numbers

OBJECTIVES:

10.1

Strong

10.2

Division

induction.

gcd(a,

and

b),

Euclidean

and

the

10.3

Linear

10.5

Representation

Before

of

sense

least

Diophantine

you

algorithms.

of

common

equations

integers

greatest

multiple,

ax

in

The

+

by

=

different

common

lcm(a,

b),

of

divisor ,

integers

a

and

b

c

bases.

start

3

1

Prove

statements

directly

by

factorization,

1

a

Show

that

n

2

e.g.

n

+

9n

+

20

+

n

−

n

is

divisible

by

6

−

n

is

divisible

by

30

+

is

an

even

number

for

all

for

all

n

∈ Z

2

∈ Z

since

n

+

9n

+

20

≡

(n

+

4)(n

+

5) 5

b

which

is

a

product

of

two

Show

that

n

consecutive +

for numbers.

For

any

two

consecutive

all

n

∈ Z

integers,

2

one

is

of

the

them

product

positive



must

Prove

of

integer

an

e.g.

even,

odd

making

statements

induction,

be

and

it

using

Prove

and n

an

+

9n

+

20

even

even.

mathematical

the

2

a

statement

Prove

the

question

two

1

statements

using

the

in

principle

of

n

i

:

P



n

n 1

i (2

)



2



(n



1)



mathematical

induction.

2

i 1

b

Proof:

When

n

=

Using

the

principle

of

mathematical

1, induction

prove

that

1

LHS

=

1

×

2

=

2,

RHS

=

2

+

0

=

2. n

n (n

Therefore

P

is

tr ue.



1

Assume

that

P

is

tr ue

for

some

k

≥



1)( 2

i 1

1.

k k

i

i.e.

i (2



k 1

)



2



(k



1)



2

i 1

When

n

=

k

+

1,

k 1

i



k 1

i (2

)



2



(k



1)



2

k 1



(k

)

i 1

k+1

=

2

+

2

k+1

(k

−

1

+

k

+

1)

=

2

+

2k (2

)

k+2

=

2

⇒

+

k

P

×

is

2

tr ue.

k+1

Since

P

is

tr ue

and

it

was

shown

that

if

P

1

is

tr ue

k

then

P

is

also

tr ue,

it

follows

by

k+1

the

principle

of

mathematical

induction



that

P

is

true

for

all n



n

2

Making

sense

of

numbers

1 ,

n

 

.

Q.E.D .

i (i



2)



1)( 2n

 6



7)

A

brief

One

journey

would

numbers

that

because

tribal

have

people

difficulty

to

number

in

the

is

think

limited,

what

we

makes

Throughout

track

base

dates

and

of

60

they

had

are

are

all

a

Here

the

not

to

much

However,

to

have

count.

a

lear n

the

number

ver y

the

skill

quantity

different

to

4.

from

although

how

good

However,

developed

beyond

not

able

people

The

3100

them

beings

ability

have

dierent

our

count

systems

sense

studies

of

show

counting

When

other

it

comes

species

number

and

of

sense

this

is

different.

back

allowing

are

our

who

histor y

to

human

kingdom.

us

far

back

of

we

quantity .

as

that

discer ning

sense

animal

through

as

devised

Mesopotamians

3400

BC.

The

special

to

have

and

up

the

Egyptians

symbol

count

Egyptian

BC

to

for

one

symbols

the

systems

had

a

aid

number

Egyptian

used

to

base

different

keeping

system

number

10

in

their

powers

using

system

of

system

10,

million.

for

the

powers

of

ten

from

10

to

one

million.

10

In

100

Europe,

system.

was

1000

Roman

One

of

developed

the

10 000

100 000

numbers

most

around

were

ancient

400

AD,

used

1

million

before

systems

is

our

the

appoximately

current

Mayan

1000

number

system

years

ahead

which

of

Chapter

1

3

European

The

counter par ts.

picture

1

below

4

But

as

it

a

6

was

in

used

The

some

numbers,

Rules

of

debt

A

for tune

the

which

about

Until

made

and

base

in

their

in

the

was

20

first

then,

wester n

arithmetic

with

came

system.

Mayan

system.

23

introduced

revolutionized

Brahmagupta

operations

numbers

20

numbers

zero

zero

indirectly

later.

mathematician

rules

the

17

number

and

centuries

negative

minus

zero

minus

minus

debt

that

used

of

ver y

wester n

arithmetic

cumbersome.

up

positive

zero.

Brahmagupta

A

Zero

some

11

number

numerals

Indian

with

A

India

many

Roman

Mayans

9

conceptual

arithmetic

The

illustrates

zero

is

is

subtracted

of

a

zero

a

debt.

is

for tune.

zero.

from

zero

a

zero

is

multiplied

a

for tune.

The

product

The

product

of

zero

by

The

product

or

quotient

of

two

for tunes

The

product

or

quotient

of

two

debts

The

product

or

quotient

of

a

debt

The

product

or

quotient

of

a

for tune

multiplied

a

by

debt

zero

or

is

is

and

for tune

is

one

one

a

is

zero.

zero.

for tune.

for tune.

for tune

and

a

debt

is

a

debt.

is

a

debt.

Investigation

Work

34

×

out

11

What

Does

do

by

Repeat

discrete

focus

will

23

from

make

your

or

be

above

for

we

to

29

results?

×

11

Work

out

some

more

products

with

the

product

numbers,

4

par ticularly

the

always

and

are

that

of

three-digit

numbers

by

111

and

explain

your

bases

interested

that

var y

belong

at

different

number

base

to

another,

as

Making

sense

of

well

bases

to

only

in

values

continuously .

the

as

numbers

integers 

and

doing

lear n

some

and

11?

works.

multiplication

positive

look

two-digit

etc...?

integers ,

that

var y

Hence,

and

the

subsets

+

thereof,

of

conjecture.

values

variables

11

five-digit

systems

opposed

on

a

conjecture

mathematics

as

×

your

work

four-digit

Number

discretely ,

and

steps

products:

11

notice

11

why

the

×

conjecture

about

Explain

In

you

your

What

following

71

numbers

1.1

the

.

In

how

this

to

section

conver t

elementar y

you

from

arithmetic.

will

one

results.

Since

the

of

a

ver y

digits

a

0

number

depends

This

is

on

age

based

4

it

a

placed

the

table

2

coefficients

of

that

can

=

2

×

+

3

4

i.e.

f

We

(x)

are

=

+

by

toes

used

and

–

only

toes.

four’

–

in

In

inter nal

the

The

a

the

the

10

as

between

×

⇒

of

of

the

a

0

polynomial

and

9

as

a

digit

number.

+

4

×

in

10

with

follows:

1

10

ve

in

is

a

0

10

+

7

×

10

for

a

toe.

20

of

for

4

toes.

the

(base

processing

system,

and

as

foot

4’

The

from

can

ngers

the

foot

number

19.

2),

for

system

of

and

three’

‘rst

the

(base

binar y

used

hand

as

be

systems

that

was

Egyptians

also

directly

output

locations

calculations

ngers

ngers

would

binar y

number

all

‘second

three

translate

and

our

whereas

Greenlanders

use

data.

do

fingers.

system,

‘second

input

storage

10

translates

would

often

mathematics

Mayan

hand

16)

of

the

10

Native

rst

So

we

have

number

electronics

representing

memor y

we

eight

the

that

base

In

system.

Greenlandic

digital

the

system

base

word

and

23 047

because

10

hexadecimal

for

=

reason

or

ngers

in

(10)

this

nger

14

and

f

the

base

ngers

and

and

7

Greenlandic

and

useful

components

0

using

resulting

systems

8)

+

to

number

all

storage

especially

10

2

+

probably ,

hence

count

meaning

(base

value

representation

base

values

4x

means

used,

fact

you

hand.

in

civilizations

Latin

were

Computer

octal

+

Most

ngers,

meaning

second

3x

many

Digitus

the

value

7

10

accustomed

mechanically .

adopted

where

with

the

3

2x

so

×

system

0

3

10

10

understand

10

number

take

base

to

below .

4

4

23 047

the

the

how

system

1

this

using

taught

in

10

0

think

been

were

decimal

10

3

can

have

Y ou

is

in

3

10

2

you

9.

on

where

illustrated

10

We

early

through

is

computer

be

in

only

Base

one

of

two

states,

on

or

off.

This

system

uses

only

two

either

1

and

0.

The

table

below

illustrates

an

example

of

how

ten

is

called

digits:

to

decimal

or

understand denar y.

the

value

4

of

a

binar y

3

2

2

2

1

1

2

0

Once

number

more

we

2

1

can

denar y .

0

2

0

in

1

think

of

the

number

as

a

polynomial

in

2

2 ×

with

coefficients

that

can

4

10 011

=

1

×

=

16

=

19

take

values

3

2

+

0

×

2

0

or

2

+

0

×

2

1

as

follows:

1

+

1

×

2

0

+

1

×

2

2

1

+

2

+

1

10

In

the

as

synthetic

core

number

in

book

we

introduced

division.

base

The

table

Hor ner’s

also

algorithm,

helps

us

to

also

calculate

known

the

10.

Chapter

1

5

Denition

A

positive

integer

N

in

base

b

notation

is

represented

by

+

N

=

(d

d

n

The

d

n−1

value

...

d

n−2

of

d

1

N

in

)

0

=

d

×

b

+

d

×

hexadecimal

=

d

is

a

×

b

∈ 

+

≤

d

+

system

N

in

d

×