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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 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) 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

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Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.