Mathematics for the international student : mathematics HL (option) : discrete mathematics, HL topic 10, FM topic 6, for use with IB diploma programme

240 Pages·2012·5.42 MB·English

by  Catherine Quinn

HAESE MATHEMATICS
Specialists in mathematics publishing
Mathematics
for the international student
Mathematics HL (Option):
Discrete Mathematics
HL Topic 10
FM Topic 6
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for use with
IB Diploma
Programme
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MATHEMATICSFORTHEINTERNATIONALSTUDENT
MathematicsHL(Option):DiscreteMathematics
CatherineQuinn B.Sc.(Hons),Grad.Dip.Ed.,Ph.D.
PeterBlythe B.Sc.
ChrisSangwin M.A.,M.Sc.,Ph.D.
RobertHaese B.Sc.
MichaelHaese B.Sc.(Hons.),Ph.D.
HaeseMathematics
152RichmondRoad,Marleston, SA5033,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haesemathematics.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-921972-34-8
©Haese&HarrisPublications2014
PublishedbyHaeseMathematics.
152RichmondRoad,Marleston, SA5033,AUSTRALIA
FirstEdition 2014
ArtworkbyBrianHouston.
CoverdesignbyPiotrPoturaj.
TypesetinAustraliabyDeanneGallasch.TypesetinTimesRoman10.
ComputersoftwarebyTimLee.
ProductionworkbyAnnaRijkenandMarkHumphries.
PrintedinChinabyProlongPressLimited.
The textbook and its accompanying CD have been developed independently of the International
Baccalaureate Organization (IBO). The textbook and CD are in no way connected with, or endorsed by,
theIBO.
This book is copyright. Except as permitted by the CopyrightAct (any fair dealing for the purposes of
private study, research, criticism or review), no part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese
Mathematics.
Copying foreducational purposes:Where copies of part or the whole of the book are made under Part
VB of the CopyrightAct, the law requires that the educational institution or the body that administers it
has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the
CopyrightAgencyLimited.
Acknowledgements: While every attempt has been made to trace and acknowledge copyright, the authors
and publishers apologise for any accidental infringement where copyright has proved untraceable. They
wouldbepleasedtocometoasuitableagreementwiththerightfulowner.
Disclaimer:Alltheinternetaddresses(URLs)giveninthisbookwerevalidatthetimeofprinting.While
the authors and publisher regret any inconvenience that changes of address may cause readers, no
responsibilityforanysuchchangescanbeacceptedbyeithertheauthorsorthepublisher.
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FOREWORD
MathematicsHL(Option):DiscreteMathematicshasbeenwrittenasacompanionbooktothe
MathematicsHL(Core)textbook.Together,theyaimtoprovidestudentsandteacherswith
appropriatecoverageofthetwo-yearMathematicsHLCourse,tobefirstexaminedin2014.
Thisbookcoversallsub-topicssetoutinMathematicsHLOptionTopic10andFurtherMathematics
HLTopic6,DiscreteMathematics.
Theaimofthistopicistointroducestudentstothebasicconcepts,techniquesandmainresultsin
numbertheoryandgraphtheory.
Detailedexplanationsandkeyfactsarehighlightedthroughoutthetext.Eachsub-topiccontains
numerousWorkedExamples,highlightingeachstepnecessarytoreachtheanswerforthatexample.
TheoryofKnowledgeisacorerequirementintheInternationalBaccalaureateDiplomaProgramme,
wherebystudentsareencouragedtothinkcriticallyandchallengetheassumptionsofknowledge.
DiscussiontopicsforTheoryofKnowledgehavebeenincludedonpages140and160.Theseaimto
helpstudentsdiscoverandexpresstheirviewsonknowledgeissues.
TheaccompanyingstudentCDincludesaPDFofthefulltextandaccesstospeciallydesigned
softwareandprintablepages.
GraphicscalculatorinstructionsforCasiofx-9860GPlus,Casiofx-CG20,TI-84PlusandTI-nspire
areavailablefromiconsinthebook.
Fullyworkedsolutionsareprovidedatthebackofthetext,howeverstudentsareencouragedto
attempteachquestionbeforereferringtothesolution.
Itisnotourintentiontodefinethecourse.Teachersareencouragedtouseotherresources.Wehave
developedthisbookindependentlyoftheInternationalBaccalaureateOrganization(IBO)in
consultationwithexperiencedteachersofIBMathematics.ThetextisnotendorsedbytheIBO.
Inthischangingworldofmathematicseducation,webelievethatthecontextualapproachshownin
thisbook,withassociateduseoftechnology,willenhancethestudentsunderstanding,knowledge
andappreciationofmathematicsanditsuniversalapplications.
Wewelcomeyourfeedback.
Email: [email protected] CTQ PJB CS
Web: www.haesemathematics.com.au RCH PMH
ACKNOWLEDGEMENTS
The authors and publishers would like to thank all those teachers who offered advice and
encouragementonthisbook.
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USING THE INTERACTIVE STUDENT CD
TheinteractiveCDisidealforindependentstudy.
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Santuddpernatcsticcaen.TrehveisCitDcoanlscoephtsastatuhgehttexintocflatshseabnodoukn,daellrotawkiengthsetiurdoewntnsrtoevliesaiovne HHLLOOppttiioonnwww.haesemathematics.com.auMMaaDDtthhiisseeccmmrreeaatt©eettiicc20ss14
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thetextbookatschoolandkeeptheCDathome. ffoorruuDDsseeiiwwsscciittrrhheeIIttBBeeDDMMiippaallttoohhmmeeaammPPrraaoottggiirrccaassmmmmee
HHaaeesseeMMaatthheemmaattiiccss
By clicking on the relevant icon, a range of interactive features can be
accessed:
INTERACTIVE
(cid:2) Graphics calculator instructions for the Casio fx-9860G Plus, LINK
Casiofx-CG20,TI-84PlusandtheTI-nspire
(cid:2) Interactivelinkstosoftware
(cid:2) Printablepages
GRAPHICS
CALCULATOR
INSTRUCTIONS
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TABLEOFCONTENTS 5
TABLE OF CONTENTS
SYMBOLS AND NOTATION USED IN THIS BOOK 6
1 NUMBER THEORY 9
A Mathematicalinduction 13
B Recurrencerelations 18
C Divisibility,primenumbers,andtheDivisionAlgorithm 41
D GCD,LCM,andtheEuclideanAlgorithm 49
E Primenumbers 62
F Congruences 66
G TheChineseRemainderTheorem 75
H Divisibilitytests 79
I Fermat’sLittleTheorem 82
J ThePigeonholePrinciple(Dirichlet’sPrinciple) 86
2 GRAPH THEORY 89
A Terminology 91
B Fundamentalresultsofgraphtheory 97
C Journeysongraphs 100
D Planargraphs 110
E Treesandalgorithms 117
F TheChinesePostmanProblem(CPP) 128
G TheTravellingSalesmanProblem(TSP) 132
THEORYOFKNOWLEDGE(NPproblems) 140
ReviewsetA 142
ReviewsetB 145
ReviewsetC 148
APPENDIX(Methodsofproof) 151
THEORYOFKNOWLEDGE(AxiomsandOccam’srazor) 160
WORKEDSOLUTIONS 161
INDEX 236
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6
SYMBOLS AND NOTATION USED IN THIS BOOK
¼ is approximatelyequal to
> is greater than
> is greater than or equal to
< is less than
6 is less than or equal to
f......g the set of all elements......
fx , x , ....g the set with elements x , x , ....
1 2 1 2
2 is an element of
2= is not an element of
N the set of all natural numbers f0, 1, 2, 3, ....g
Z the set of integers f0, §1, §2, §3, ....g
Z+ the set of positive integers f1, 2, 3, ....g
R the set of real numbers
[ union
\ intersection
Zm the set of equivalenceclasses f0, 1, 2, ...., m¡1g of integers modulo m
) implies that
)Á does not imply that
, if and only if
f(x) the image of x under the functionf
Pn
ui u1+u2+u3+::::+un
i=1
ajb a divides b
gcd(a, b) the greatestcommondivisor of a and b
lcm(a, b) the least commonmultiple of a and b
a´b(modm) a is congruentto b modulo m
sin, cos, tan the circularfunctions
arcsin, arccos, arctan the inverse circular functions
cis µ cosµ+isinµ
n! n£(n¡1)£(n¡2)£::::£3£2£1
¡ ¢
n!
n
r r!(n¡r)!
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7
(ak::::a2a1a0) digital form of the integer ak10k+::::+a2102+a110+a0
(ak::::a2a1a0)n digital form of the integer aknk+::::+a2n2+a1n+a0
Pn a propositiondefined for some n
fn the nth term of the Fibonaccisequence
deg(A) the degree of vertex A
0
G the complementof graph G
Kn the complete graph on n vertices
Km,n the complete bipartitegraph with m vertices in one set and n in the other
Cn the cycle graph on n vertices
Wn the wheel graph on n vertices
deg(F) the degree of face F
wt(T) the weight of tree T
wt(VW) the weight of edge VW
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1
Number theory
Contents:
A Mathematical induction
B Recurrence relations
C Divisibility, prime numbers, and the Division Algorithm
D GCD, LCM, and the Euclidean Algorithm
E Prime numbers
F Congruences
G The Chinese Remainder Theorem
H Divisibility tests
I Fermat’s Little Theorem
J The Pigeonhole Principle (Dirichlet’s Principle)
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10 NUMBERTHEORY (Chapter1)
INTRODUCTION TO NUMBER THEORY
You might think that integers are the simplest of mathematical objects. However, their properties lead
to some very deep and satisfyingmathematics. The study of the propertiesof integersis callednumber
theory.
In this course we will study:
² techniquesof proof
² applicationsof algorithms, which are methods of mathematicalreasoning and solution
² a developmentof the number system with modulararithmetic
² the use and proof of important theorems.
SETS OF INTEGERS
The set of all integersis Z = f0, §1, §2, §3, §4, §5, ....g.
The set of all positiveintegers is Z+ = f1, 2, 3, 4, 5, ....g.
The set of natural numbers is N = f0, 1, 2, 3, ....g = Z+ [ f0g.
NOTATION
2 reads is in or is an element of or is a memberof
) reads implies
, reads if and only if
ajb reads a divides b or a is a factor of b fajb)b=na for some n2Zg.
gcd(a, b) reads the greatest commondivisor of a and b, which is the highest commonfactor
of a and b
lcm(a, b) reads the least commonmultipleof a and b.
(ak::::a2a1a0) is the digital form of the integer ak10k+::::+a2102+a110+a0
(ak::::a2a1a0)n is the digital form of the integer aknk+::::+a2n2+a1n+a0
If the digits are all known then we leave off the brackets.
For example, 101101 =1£25+0£24+1£23+1£22+0£21+1£20.
2
PRIME AND COMPOSITE INTEGERS
A positiveinteger p is prime if p>1 and the only factors of p are 1 and p itself.
If a positive integer m, m>1, is not prime, it is called composite.
The integer 1 is neither prime nor composite.
For example:
² 2, 3, 5, 7, 11 are prime numbers.
² 1, 4, 6, 9 are not prime numbers. In particular, 4 = 2£2, 6 = 2£3, and 9 = 3£3 are
examplesof compositenumbers.
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