Proof patterns

189 Pages·2015·2.981 MB·English

by  Mark Joshi

Mark Joshi
Proof
Patterns
Proof Patterns
Mark Joshi
Proof Patterns
123
MarkJoshi
Centre forActuarial Studies
Universityof Melbourne
Melbourne,VIC
Australia
ISBN 978-3-319-16249-2 ISBN 978-3-319-16250-8 (eBook)
DOI 10.1007/978-3-319-16250-8
LibraryofCongressControlNumber:2015932521
MathematicsSubjectClassification:00–01,97D50
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Preface
Patterns have become a common theme in many fields of academic study. In
programming, in particular, the book “Design Patterns” has become highly influ-
ential and it is now customary to discuss programs in terms of the patterns used.
Theprogrammersgenerallyattributetheideaofadesignpatterntoarchitecture.The
fundamental idea is that each field has a collection of ways of breaking down
problems into component pieces. Understanding these methodologies explicitly
then leads to greater comprehension, facilitates learning and simplifies problem
solving. Rather than attempting a problem cold, one first sees whether known
patterns work. Even if they all fail, understanding why they fail defines the
problem.
Inthisbook,myobjectiveistoidentifyandteachmanyofthecommonpatterns
thatariseinpuremathematics.Icallthese“ProofPatterns”.Themainoriginalityin
the presentation is that examples are focussed about each pattern and drawn from
differentareas.Thisdiffersfromtheusualstyleofteachingpuremathematicswhere
a topic is chosen and dissected; patterns are then drawn in as needed, and they are
oftennotexplicitlymentioned.Afterstudyingenoughtopicsthelearnerpicksupa
varietyofpatterns,andthedifficultyofstudyinganewareaisoftendeterminedby
the degree of unfamiliarity with its patterns.
Thisbookisintendedtodoavarietyofthings.Ononelevel,myobjectiveisto
teachthebasicpatterns.Onanother,itisintendedasatasterforpuremathematics.
Thereaderwillgainalittleknowledge onavarietyoftopicsandhopefullylearna
littleaboutwhatpuremathematicsis.Onathirdlevel,theintentionofthebookisto
make a case for the explicit recognition of patterns when teaching pure mathe-
matics. On a fourth level, it is simply an enjoyable romp through topics I love.
One powerful tool of pure mathematics which I intentionally avoid is that of
abstraction. I believe that patterns and concepts are best learnt via the study of
concrete objects wherever possible. Whilst one must go abstract eventually to
obtain the full power and generality of results, a proof or pattern that has already
been understood in a concrete setting is much easier to comprehend and apply.
v
vi Preface
A side effect of this avoidance is that the patterns are not formally defined since
such definitions would require a great deal of abstraction.
Thetargetreaderofthisbookwillalreadybefamiliarwiththeconceptofproof
but need not know much more. So whilst I assume very few results from pure
mathematics,thereaderwhodoesnotknowwhataproofiswillstruggle.Thereare
several excellent texts such as Eccles, Velleman and Houston for such readers to
study before reading here. In particular, I regard this book as a second book on
proof, and my hope is that the reader will find that the approach here eases their
study of many areas of pure mathematics.
Itrytobuildupeverything fromthegroundupasmuchaspossible.Itherefore
try to avoid the “pull a big theorem out of the hat” style of mathematics presen-
tation.Theemphasisismuchmoreonhowtoproveresultsratherthanontryingto
impress with theorems whose proofs are far beyond the book’s scope. I do occa-
sionally use concepts from analysis before these are formally defined such as a
convergent sequence. Hopefully, the reader who has not studied analysis will be
able to work with their intuitive notions of these objects.
Inevitably,aswithmanyintroductorybooksonproof,manyexamplesaredrawn
from combinatorics and elementarynumbertheory. Thisreflects thefact that these
areas require fewer prerequisites than most and so patterns can be discussed in
simplesettings.However,Ialsodrawonavarietyofareasincludinggrouptheory,
linear algebra, computer science, analysis, topology, Euclidean geometry, and set
theory to emphasize patterns’ universality.
There is little if any originality in the mathematical results in this book: the
objectivewastoprovideadifferentpresentationratherthannewresults.Welookat
the “Four-Colour problem” at various points. Our treatment is very much inspired
byRobinWilson’sexcellentbook“Fourcolourssuffice”andIrecommendittoany
readerwhoseinteresthasbeenpiqued.Abookwithsomesimilaritiestothisonebut
requiring a little more knowledge from the reader is “Proofs from the BOOK” by
AignerandZiegler.Theemphasisthereismoreonbeautyinproofthanonpatterns
anditisagoodfollowonforthereaderwhowantsmore.However,Idohopethat
any reader of this book will develop some appreciation for the beauty of
mathematics.
Manyofthepatternsinthisbookhavenotbeennamedbeforealthoughtheyare
in widespread use. I have therefore invented their names. I hope that these new
names will prove popular. I apologise to those who dislike them.
For the reader who has forgotten or never knew mathematical terminology,
I have included a glossary in Appendix A. This also includes definitions of the
standard sets of numbers. For clarity, let me say right here that in this book 0 is a
naturalnumber.ThisisthewayIwastaughtasanundergraduateanditistoofirmly
embedded in my psyche for me to use any other definition. The term counting
numbers denoted N will be used for the natural numbers excluding zero.
1
Thisbookisultimatelyanexpressionofmyphilosophyofhowtoapproachthe
teaching of mathematics. My views have been shaped by interactions with
Preface vii
innumerable former teachers, students and colleagues and I thank them all.
I particularly thank Alan Beardon and Navin Ranasinghe for their detailed com-
ments on a former version of the text. I also thank some anonymous referees for
their constructive comments.
Melbourne, 2014 Mark Joshi
Contents
1 Induction and Complete Induction . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Examples of Induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Why Does Induction Hold? . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Induction and Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Triangulating Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Double Counting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Summing Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Vandermonde’s Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Fermat’s Little Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Icosahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Pythagoras’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 The Pigeonhole Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Rationals and Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Lossless Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 More Irrationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Divisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Division and Well-Ordering . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Algorithms and Highest Common Factors . . . . . . . . . . . . . . 26
4.4 Lowest Terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
ix
x Contents
4.5 Euclid’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 The Uniqueness of Prime Decompositions . . . . . . . . . . . . . . 30
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Contrapositive and Contradiction . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 An Irrational Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 The Infinitude of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 More Irrationalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5 The Irrationality of e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.6 Which to Prefer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.7 Contrapositives and Converses . . . . . . . . . . . . . . . . . . . . . . 40
5.8 The Law of the Excluded Middle . . . . . . . . . . . . . . . . . . . . 40
5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Intersection-Enclosure and Generation. . . . . . . . . . . . . . . . . . . . . 43
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Examples of Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 Advanced Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.4 The Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.5 Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.6 Fields and Square Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Difference of Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Dominoes and Triminoes. . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.4 Cardinality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.5 Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.6 Divisibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8 Linear Dependence, Fields and Transcendence. . . . . . . . . . . . . . . 65
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.2 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.3 Linear Dependence and Algebraic Numbers . . . . . . . . . . . . . 69
8.4 Square Roots and Algebraic Numbers . . . . . . . . . . . . . . . . . 70
8.5 Transcendental Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Contents xi
9 Formal Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 Ruler and Compass Constructions. . . . . . . . . . . . . . . . . . . . 73
9.3 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
10 Equivalence Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10.2 Constructing the Integers . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10.3 Constructing the Rationals . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.4 The Inadequacy of the Rationals. . . . . . . . . . . . . . . . . . . . . 88
10.5 Constructing the Reals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
10.6 Convergence of Monotone Sequences . . . . . . . . . . . . . . . . . 93
10.7 Existence of Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . 94
10.8 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
10.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
11 Proof by Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.2 Co-prime Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.3 Classifying Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . 98
11.4 The Non-existence of Pythagorean Fourth Powers. . . . . . . . . 101
11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
12 Specific-generality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
12.2 Reducing the Fermat Theorem . . . . . . . . . . . . . . . . . . . . . . 105
12.3 The Four-Colour Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 106
12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
13 Diagonal Tricks and Cardinality. . . . . . . . . . . . . . . . . . . . . . . . . 109
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
13.2 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
13.3 Infinite Sets of the Same Size. . . . . . . . . . . . . . . . . . . . . . . 110
13.4 Diagonals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
13.5 Transcendentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
13.6 Proving the Schröder–Bernstein Theorem. . . . . . . . . . . . . . . 116
13.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
14 Connectedness and the Jordan Curve Theorem . . . . . . . . . . . . . . 119
14.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
14.2 Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
14.3 The Jordan Closed-Curve Theorem . . . . . . . . . . . . . . . . . . . 122
14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125