Table Of ContentExploring Mathematics
An Engaging Introduction to Proof
Exploring Mathematics gives students experience with doing mathematics - interrogating
mathematical claims, exploring definitions, forming conjectures, attempting proofs, and
presenting results - and engages them with examples, exercises, and projects that pique their
curiosity. Written with a minimal number of pre-requisites, this text can be used by college
students in their first and second years of study and by independent readers who want an
accessible introduction to theoretical mathematics. Core topics include proof techniques,
sets, functions, relations, and cardinality, with selected additional topics that provide many
possibilities for further exploration.
With a problem-based approach to investigating the material, students develop interesting
examples and theorems through numerous exercises and projects. In-text exercises, with
complete solutions or robust hints included in an appendix, help students explore and master
the topics being presented. The end-of-chapter exercises and projects provide students
opportunities to confirm their understanding of core material, learn new concepts, and
develop mathematical creativity.
John Meier is the David M. ’70 and Linda Roth Professor of Mathematics at Lafayette
College, where he also served as Dean of the Curriculum. His research focuses on geometric
group theory and involves algorithmic, combinatorial, geometric, and topological issues
that arise in the study of infinite groups. In addition to teaching awards from Cornell
University and Lafayette College, Professor Meier is the proud recipient of the James
Crawford Teaching Prize from the Eastern Pennsylvania and Delaware section of the
Mathematical Association of America.
Derek Smith is Associate Professor of Mathematics at Lafayette College. His research
focuses on algebra, combinatorics, and geometry.
He has taught a wide variety of undergraduate courses in mathematics and other subjects in
both the United States and Europe. He is the recipient of multiple teaching awards at
Lafayette, and his work has been supported by the Mathematical Association of America
and the National Science Foundation. Professor Smith is a former editor of the problem
section of Math Hor'izons.
CAMBRIDGE MATHEMATICAL TEXTBOOKS
Cambridge Mathematical Textbooks is a program of undergraduate and beginning
graduate level textbooks for core courses, new courses, and interdisciplinary courses in pure
and applied mathematics. These texts provide motivation with plenty of exercises of
varying difficulty, interesting examples, modern applications, and unique approaches to the
material.
Advisory Board
John B. Conway, George Washington University
Gregory’ F. Lawler, University of Chicago
John M. Lee, University of Washington
John Meier, Lafayette College
Lawrence C. Washington, University of Maryland, College Park
A complete list of books in the series can be found at
http://www.cambridge.org/mathematics
Recent titles include the following:
Chance, Strategy, and Choice: An Introduction to the Mathematics of Games and Elections,
S. B. Smith
Set Theory: A First Course, D. W. Cunningham
Chaotic Dynamics: Fractals, Tilings, and Substitutions, G. R. Goodson
Introduction to Experimental Mathematics, S. Eilers & R. Johansen
A Second Course in Linear Algebra, S. R. Garcia & R. A. Horn
Exploring Mathematics: An Engaging Introduction to Proof, J. Meier & D. Smith
A First Course in Analysis, J. B. Conway
Exploring Mathematics
An Engaging Introduction to Proof
JOHN MEIER
Lafayette College, PA, USA
DEREK SMITH
Lafayette College, PA, USA
University Printing House, Cambridge CB2 8BS, United Kingdom
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Cambridge University Press is part of the University of Cambridge.
It furthers the University's mission by disseminating knowledge in the pursuit of
education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridgc.org/9781107128989
© John Meier and Derek Smith 2017
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2017
Printed in the United States of America by Sheridan Books, Inc.
A catalogue record for this publication is available from the British Library.
ISBN 978-1-107-12898-9 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
Preface page xi
1 Let’s Play! 1
1.1 A Direct Approach 1
1.2 Fibonacci Numbers and the Golden Ratio 3
1.3 Inductive Reasoning 6
1.4 Natural Numbers and Divisibility 9
1.5 The Primes 10
1.6 The Integers 11
1.7 The Rationals, the Reals, and the Square Root of 2 13
1.8 End-of-Chapter Exercises 15
2 Discovering and Presenting Mathematics 25
2.1 Truth, Tabulated 25
2.2 Valid Arguments and Direct Proofs 29
2.3 Proofs by Contradiction 32
2.4 Converse and Contrapositive 34
2.5 Quantifiers 35
2.6 Induction 32
2.7 Ubiquitous Terminology 44
2.8 The Process of Doing Mathematics 45
2.9 Writing Up Your Mathematics 50
2.10 End-of-Chapter Exercises 55
3 Sets 68
3.1 Set Builder Notation 68
3.2 Sizes and Subsets 69
3.3 Union, Intersection, Difference, and Complement 71
3.4 Many Laws and a Few Proofs 73
3.5 Indexing 75
3.6 Cartesian Product 72
3.7 Power 78
3.8 Counting Subsets 81
3.9 A Curious Set 83
3.10 End-of-Chapter Exercises 85
4 The Integers and the Fundamental Theorem of Arithmetic 94
4.1 The Well-Ordering Principle and Criminals 94
4.2 Integer Combinations and Relatively Prime Integers 96
4.3 The Fundamental Theorem of Arithmetic 98
vii
viii Contents
4.4 LCM and GCD 100
4.5 Numbers and Closure 102
4.6 End-of-Chapter Exercises 106
5 Functions 1 1 1
5.1 What is a Function? 1 1 1
5.2 Domain, Codomain and Range 114
5.3 Injective, Surjective, and Bijective 115
5.4 Composition 118
5.5 What is a Function? Redux! 1 2 0
5.6 Inverse Functions 1 2 2
5.7 Functions and Subsets 1 2 5
5.8 A Few Facts About Functions and Subsets 129
5.9 End-of-Chapter Exercises 131
6 Relations 141
6.1 Introduction to Relations 141
6.2 Partial Orders 142
6.3 Equivalence Relations 145
6.4 Modulo m 148
6.5 Modular Arithmetic 149
6.6 Invertible Elements 152
6.7 End-of-Chapter Exercises 156
7 Cardinality 164
7.1 The Hilbert Hotel, Count von Count, and Cookie Monster 164
7.2 Cardinality 166
7.3 Countability 1 6 8
7.4 Key Countability Lemmas 169
7.5 Not Every Set is Countable 172
7.6 Using the Schröder-Bernstein Theorem 1 7 5
7.7 End-of-Chapter Exercises 178
8 The Real Numbers 1 8 4
8.1 C o m p l e t e n e s s 184
8.2 The Archimedean Property 1 8 6
8.3 Sequences of Real Numbers 1 8 8
8.4 Geometric Series 191
8.5 The Monotone Convergence Theorem 195
8.6 Famous Irrationals 1 2 7
8.7 End-of-Chapter Exercises 203
9 Probability and Randomness 2 0 9
9.1 A Class of Lyin' Weasels 209
9.2 Probability 2 1 0
9.3 Revisiting Combinations 214
Contents ix
9.4 Events and Random Variables 216
9.5 Expected Value 217
9.6 Flipped or Faked? 218
9.7 End-of-Chapter Exercises 222
10 Algebra and Symmetry 229
10.1 An Example from Modular Arithmetic 229
10.2 The Symmetries of a Square 230
10.3 Group Theory 234
10.4 Cayley Tables 236
10.5 Group Properties 238
10.6 Isomorphism 240
10.7 Isomorphism and Group Properties 241
10.8 Examples of Isomorphic and Non-isomorphic Groups 243
10.9 End-of-Chapter Exercises 246
11 Projects 250
11.1 The Pythagorean Theorem 250
11.2 Chomp and the Divisor Game 253
11.3 Arithmetic-Geometric Mean Inequality 256
11.4 Complex Numbers and the Gaussian Integers 258
11.5 Pigeons! 262
11.6 Mirsky’s Theorem 264
11.7 Euler’s Totient Function 268
11.8 Proving the Schröder-Bernstein Theorem 271
11.9 Cauchy Sequences and the Real Numbers 272
11.10 The Cantor Set 277
11.11 Five Groups of Order 8 280
Solutions, Answers, or Hints to In-Text Exercises 282
Bibliography 318
Index 321
Preface
Mathematics is a fascinating discipline that calls for creativity, imagination, and the
mastery of rigorous standards of proof. This book introduces students to these facets
of the field in a problem-focused setting. For over a decade, we and many others
haveuseddraftchapters ofExploringMathematics astheprimarytextforLafayette’s
Transition to Theoretical Mathematics course. Our collective experience shows that
this approach assists students in their transition from primarily computational classes
towardmoreadvancedmathematics,anditencouragesthemtocontinuealongthispath
by demonstrating that while mathematics can at times be challenging, it is also very
enjoyable.
HerearesomeofthekeyfeaturesofExploringMathematics.
• The sections are short, and core topics are covered in chapters that present impor-
tant material with minimal pre-requisites. This structure provides flexibility to the
instructorintermsofpacingandcoverage.
• Mathematicalmaturityrequiresbothafacilitywithwritingproofsandcomfortwith
abstraction and creativity. We help students develop these abilities throughout the
book,beginningwiththeinitialchapters.
• A student does not learn mathematics by passive reading. It is through the creation
of examples, questioning if results can be extended, and other such in-the-margin
activities that a student learns the subject. We encourage this behavior by includ-
ingfrequentin-textexercisesthatservenotonlytocheckunderstanding,butalsoto
developmaterial.
• We construct many mathematical objects that are elementary in their definition
and commonly referenced in upper-level classes. These are woven throughout
the text, with related exercises providing numerous opportunities for independent
investigationsoftheirimportantproperties.
• Eachchapterconcludeswitharobustmixtureofexercisesrangingfromtheroutine
toratherchallengingproblems,andthebookconcludeswithacollectionofprojects:
guidedexplorationsthatstudentscanworkonindividuallyoringroups.
TheseandotherfundamentalaspectsofExploringMathematicsaredescribedingreater
detail below, where we also indicate different ways an instructor can map out the
materialthatcanbecoveredinasingleterm.
AnActiveApproach
Ourexperienceteachingcoursesthatintroducestudentstomathematicalproofsshows
thatthespiritofmathematicsiseffectivelytaughtwithafocusonproblem-solving.Itis
indoingmathematics–byexploringdefinitions,formingconjectures,andworkingon
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