Comprehensive Mathematics for Computer Scientists 1: Sets and Numbers, Graphs and Algebra, Logic and Machines, Linear Geometry

387 Pages·2006·3.9 MB·English

by  Kevin McCrimmon

Guerino Mazzola • Gérard Milmeister •
Jody Weissmann
Comprehensive Mathematics
for Computer Scientists 1
Sets and Numbers, Graphs and Algebra,
Logic and Machines, Linear Geometry
(Second Edition)
With 118 Figures
Guerino Mazzola
Gérard Milmeister
Jody Weissmann
Department of Informatics
University Zurich
Winterthurerstr. 190
8057 Zurich, Switzerland
The text has been created using LATEX2ε. The graphics were drawn using Dia, an
open-source diagramming software. The main text has been set in the Y&Y Lucida
Bright type family, the headings in Bitstream Zapf Humanist 601.
Library of Congress Control Number: 2006929906
Mathematics Subject Classification (2000): 00A06
ISBN (First Edition) 3-540-20835-6
ISBN 3-540-36873-6 Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the
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prosecution under the German Copyright Law.
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© Springer-Verlag Berlin Heidelberg 2006
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Cover Design: Erich Kirchner, Heidelberg
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Preface to the Second Edition
A second edition of a book is a success and an obligation at the same
time.Wearesatis(cid:2)edthatanumberofuniversitycourseshavebeenorga›
nizedonthebasisofthe(cid:2)rstvolumeofComprehensiveMathematicsfor
Computer Scientists. The instructors recognized that the self›contained
presentation of a broad specturm of mathematical core topics is a (cid:2)rm
point of departure for a sustainable formal education in computer sci›
ence.
We feel obliged to meet the valuable feedback of the responsible in›
structors of such courses, in particular of Joel Young (Computer Science
Department, Brown University) who has provided us with numerous re›
marks on misprints, errors, or obscurities. We would like to express our
gratitudeforthesecollaborativecontributions.Wehaverereadtheentire
text and not only eliminated identi(cid:2)ed errors, but also given some addi›
tional examples and explications to statements and proofs which were
exposedinatooshorthandstyle.
A second edition of the second volume will be published as soon as the
errata, the suggestions for improvements, and the publisher’s strategy
areinharmony.
Zurich, GuerinoMazzola
June2006 GØrardMilmeister
JodyWeissmann
Preface
The need for better formal competence as it is generated by a sound
mathematical education has been con(cid:2)rmed by recent investigations by
professional associations,butalso by IT opinionleaders suchas Niklaus
Wirth or Peter Wegner. It is rightly argued that programming skills are a
necessarybutbyfarnotsu(cid:20)cientquali(cid:2)cationfordesigningandcontrol›
lingtheconceptualarchitectureofvalidsoftware.Often,thede(cid:2)ciencyin
formalcompetenceiscompensatedbytrialanderrorprogramming.This
strategy may lead to uncontrolled code which neither formally nor ef›
fectively meets the given objectives. According to the global view such
bad engineering practice leads to massive quality breakdowns with cor›
respondingeconomicalconsequences.
Improved formal competence is also urged by the object›oriented para›
digm which progressively requires a programming style and a design
strategy of high abstraction level in conceptual engineering. In this con›
text,thearsenalofformaltoolsmustapplytocompletelydi(cid:19)erentprob›
lem situations. Moreover, the dynamics and life cycle of hard› and soft›
ware projects enforce high (cid:3)exibility of theorists and executives on all
levels of the computer science academia and IT industry. This (cid:3)exibil›
ity can only be guaranteed by a propaedeutical training in a number of
typicalstylesofmathematicalargumentation.
Withthisinmind,writinganintroductorybookonmathematicsforcom›
puter scientists is a somewhat delicate task. On the one hand, computer
science delves into the most basic machinery of human thought, such
as it is traced in the theory of Turing machines, rewriting systems and
grammars,languages,andformallogic.Ontheotherhand,numerousap›
plications of core mathematics, such as the theory of Galois (cid:2)elds (e.g.,
for coding theory), linear geometry (e.g., for computer graphics), or dif›
ferential equations (e.g., for simulation of dynamic systems) arise in any
VIII Preface
relevanttopicofcomputationalscience.Inviewofthiswide(cid:2)eldofmath›
ematical subjects the common practice is to focus one’s attention on
a particular bundle of issues and to presuppose acquaintance with the
background theory, or else to give a short summary thereof without any
furtherdetails.
Inthisbook,wehavechosenadi(cid:19)erentpresentation.Theideawastoset
forthandprovetheentirecoretheory,fromaxiomaticsettheorytonum›
bers, graphs, algebraic and logical structures, linear geometry(cid:151)in the
present (cid:2)rst volume, and then, in the second volume, topology and cal›
culus, di(cid:19)erential equations, and more specialized and current subjects
such as neural networks, fractals, numerics, Fourier theory, wavelets,
probabilityandstatistics,manifolds,andcategories.
There isa pricetopay for thiscomprehensivejourneythroughtheover›
whelmingly extended landscape of mathematics: We decided to omit
any not absolutely necessary rami(cid:2)cation in mathematical theorization.
Rather it was essential to keep the global development in mind and to
avoid an unnecessarily broad approach. We have therefore limited ex›
plicit proofs to a length which is reasonable for the non›mathematician.
Inthecaseoflengthyandmoreinvolvedproofs,werefertofurtherread›
ings. For a more profound reading we included a list of references to
originalpublications.Afterall,thestudentshouldrealizeasearlyaspos›
sible in his or her career that science is vitally built upon a network of
linkstofurtherknowledgeresources.
We have, however, chosen a a modern presentation: We introduce the
language of commutative diagrams, universal properties and intuitionis›
tic logic as advanced by contemporary theoretical computer science in
its topos›theoretic aspect. This presentation serves the economy and el›
egance of abstraction so urgently requested by opinion leaders in com›
puter science. It also shows some original restatements of well›known
facts, for example in the theory of graphs or automata. In addition, our
presentation o(cid:19)ers a glimpse of the unity of science: Machines, formal
conceptarchitectures,andmathematicalstructuresareintimatelyrelated
witheachother.
Beyond a traditional (cid:147)standalone(cid:148) textbook, this text is part of a larger
formal training project hosted by the Department of Informatics at the
University of Zurich. The online counterpart of the text can be found
on http://math.ifi.unizh.ch.Ito(cid:19)ers access to this material and in›
cludesinteractivetoolsforexamplesandexercisesimplementedbyJava
Preface IX
applets and script›based dynamic HTML. Moreover, the online presenta›
tionallowsswitchingbetweentextualnavigationviaclassicallinksanda
quasi›geographical navigation on a (cid:147)landscape of knowledge(cid:148). In the lat›
ter, parts, chapters, axioms, de(cid:2)nitions, and propositions are visualized
bycontinents,nations,cities,andpaths.Thissurfacestructuredescribes
thetoplayerofathree›foldstrati(cid:2)cation(seethefollowingscreenshotof
somewindowsoftheonlineversion).
On top are the facts, below, in the middle layer, the user will (cid:2)nd the
proofs, and in the third, deepest stratum, one may access the advanced
topics, such as (cid:3)oating point arithmetic, or coding theory. The online
counterpart of the book includes two important addenda: First, a list of
X Preface
errata can be checked out. The reader is invited to submit any error en›
counteredwhilereadingthebookortheonlinepresentation.Second,the
subjectspectrum,beitintheory,examples,orexercises,isconstantlyup›
datedandcompletedand,ifappropriate,extended.Itisthereforerecom›
mended and bene(cid:2)cial to work with both, the book and its online coun›
terpart.
This book is a result of an educational project of the E›Learning Center
of the University of Zurich. Its production was supported by the Depart›
ment of Informatics, whose infrastructure we were allowed to use. We
would like to express our gratitude to these supporters and hope that
the result will yield a mutual pro(cid:2)t: for the students in getting a high
quality training, and for the authors for being given the chance to study
and develop a core topic of formal education in computer science. We
also deeply appreciate the cooperation with the Springer Publishers, es›
pecially with Clemens Heine, who managed the book’s production in a
completelye(cid:20)cientandunbureaucraticway.
Zurich, GuerinoMazzola
February2004 GØrardMilmeister
JodyWeissmann
Contents
I Sets, Numbers, and Graphs 1
1 Fundamentals(cid:151)Concepts andLogic 3
1.1 PropositionalLogic ...................................... 4
1.2 ArchitectureofConcepts ................................ 8
2 AxiomaticSetTheory 15
2.1 TheAxioms ............................................. 17
2.2 BasicConceptsandResults.............................. 20
3 BooleanSetAlgebra 25
3.1 TheBooleanAlgebraofSubsets.......................... 25
4 FunctionsandRelations 29
4.1 GraphsandFunctions ................................... 29
4.2 Relations................................................ 41
5 OrdinalandNaturalNumbers 45
5.1 OrdinalNumbers........................................ 45
5.2 NaturalNumbers ........................................ 50
6 RecursionTheorem andUniversalProperties 55
6.1 RecursionTheorem...................................... 56
6.2 UniversalProperties ..................................... 58
6.3 UniversalPropertiesinRelationalDatabaseTheory ...... 66
7 NaturalArithmetic 73
7.1 NaturalOperations ...................................... 73
7.2 EuclidandtheNormalForms ............................ 76
8 In(cid:2)nities 79
8.1 TheDiagonalizationProcedure .......................... 79
XII Contents
9 TheClassicalNumberDomainsZ,Q,R,andC 81
9.1 IntegersZ ............................................... 82
9.2 RationalsQ ............................................. 87
9.3 RealNumbersR ......................................... 90
9.4 ComplexNumbersC .................................... 102
10 CategoriesofGraphs 107
10.1DirectedandUndirectedGraphs......................... 108
10.2MorphismsofDigraphsandGraphs ..................... 114
10.3Cycles................................................... 125
11 ConstructionofGraphs 129
12 SomeSpecialGraphs 137
12.1n›aryTrees.............................................. 137
12.2MooreGraphs ........................................... 139
13 Planarity 143
13.1Euler’sFormulaforPolyhedra ........................... 143
13.2Kuratowski’sPlanarityTheorem ......................... 147
14 FirstAdvancedTopic 149
14.1FloatingPointArithmetic ................................ 149
14.2ExampleforanAddition................................. 154
II Algebra, Formal Logic, and Linear Geometry 157
15 Monoids,Groups,Rings,andFields 159
15.1Monoids................................................. 159
15.2Groups.................................................. 163
15.3Rings.................................................... 171
15.4Fields ................................................... 177
16 Primes 181
16.1PrimeFactorization...................................... 181
16.2RootsofPolynomialsandInterpolation .................. 186
17 FormalPropositionalLogic 191
17.1Syntactics:TheLanguageofFormalPropositionalLogic .. 193
17.2Semantics:LogicalAlgebras.............................. 196
17.3Signi(cid:2)cation:Valuations ................................. 200
17.4Axiomatics .............................................. 203
Contents XIII
18 FormalPredicateLogic 209
18.1Syntactics:First›orderLanguage ......................... 211
18.2Semantics:(cid:6)›Structures.................................. 217
18.3Signi(cid:2)cation:Models..................................... 218
19 Languages, Grammars,andAutomata 223
19.1Languages............................................... 224
19.2Grammars............................................... 229
19.3AutomataandAcceptors ................................ 243
20 CategoriesofMatrixes 261
20.1WhatMatrixesAre....................................... 262
20.2StandardOperationsonMatrixes ........................ 265
20.3SquareMatrixesandtheirDeterminant .................. 271
21 ModulesandVectorSpaces 279
22 LinearDependence,Bases,andDimension 287
22.1BasesinVectorSpaces................................... 288
22.2Equations ............................................... 295
22.3A(cid:20)neHomomorphisms ................................. 296
23 AlgorithmsinLinearAlgebra 303
23.1GaussElimination ....................................... 303
23.2TheLUPDecomposition ................................. 307
24 LinearGeometry 311
24.1EuclideanVectorSpaces ................................. 311
24.2TrigonometricFunctionsfromTwo›DimensionalRotations 320
24.3Gram’sDeterminantandtheSchwarzInequality ......... 323
25 Eigenvalues,theVectorProduct,andQuaternions 327
25.1EigenvaluesandRotations............................... 327
25.2TheVectorProduct...................................... 331
25.3Quaternions............................................. 333
26 SecondAdvancedTopic 343
26.1GaloisFields ............................................ 343
26.2TheReed›Solomon(RS)ErrorCorrectionCode ........... 349
26.3TheRivest›Shamir›Adleman(RSA)EncryptionAlgorithm . 353
A FurtherReading 357