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AMS / MAA TEXTBOOKS VOL 41 Journey into VOL Discrete AMS / MAA TEXTBOOKS 41

Owen D. Byer, Deirdre . Smeltzer, and Kenneth L. Wantz Journey into Journey into Discrete Mathematics is designed for use in a fi rst course in mathematical abstraction for early-career undergraduate mathematics

majors. The important ideas of discrete mathematics are included— Owen D. Byer, Deirdre L. Smeltzer, and Kenneth L. Wantz , sets, proof writing, relations, counting, number , and —in a manner that promotes development of a mathematical mindset and prepares students for further study. While the treatment is designed to prepare the student reader for the mathematics major, the book remains attractive and appealing to students of and other problem-solving disciplines. The exposition is exquisite and engaging and features detailed descrip- tions of the thought processes that one might follow to attack the problems of mathematics. The problems are appealing and vary widely in depth and diffi culty. Careful design of the book helps the student reader learn to think like a through the exposition and the prob- lems provided. Several of the topics, including counting, , and graph theory, are visited twice: once in an introductory manner and then again in a later chapter with more advanced concepts and with a deeper perspective. Owen D. Byer and Deirdre L. Smeltzer are both Professors of Mathematics at Eastern Mennonite University. Kenneth L. Wantz is Professor of Mathematics at Regent University. Collectively the authors have special- ized expertise and research publications ranging widely over discrete mathematics and have over fi fty semesters of combined experience in teaching this subject.

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AMS/MAA TEXTBOOKS

VOL 41

Journey into Discrete Mathematics

Owen D. Byer, Deirdre L. Smeltzer, and Kenneth L. Wantz Committee on Books Jennifer J. Quinn, Chair MAA Textbooks Editorial Board Stanley E. Seltzer, Editor William Robert Green, Co-Editor Bela Bajnok Suzanne Lynne Larson Jeffrey L. Stuart Matthias Beck John Lorch Ron D. Taylor, Jr. Heather Ann Dye Michael J. McAsey Elizabeth Thoren Charles R. Hampton Virginia Noonburg Ruth Vanderpool 2010 Mathematics Subject Classification. Primary 97K20, 97K30, 97F60, 97N70, 97E30.

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Library of Congress Cataloging-in-Publication Names: Byer, Owen, author. | Smeltzer, Deirdre L., author. | Wantz, Kenneth L., 1965- author. Title: Journey into discrete mathematics / Owen D. Byer, Deirdre L. Smeltzer, Kenneth L. Wantz. Description: Providence, Rhode Island : MAA Press, an imprint of the American Mathematical Society, [2018] | : AMS/MAA textbooks ; 41 | Includes bibliographical references and index. Identifiers: LCCN 2018023584 | ISBN 9781470446963 (alk. paper) Subjects: LCSH: theory. | Number theory. | Mathematical . | Combinatorial analysis. | AMS: , graph theory, theory, – Combinatorics. msc | Mathematics education – Combinatorics, graph theory, , statistics – Graph the- ory. msc | Mathematics education – , number theory – Number theory. msc | Mathematics education – Numerical mathematics – Discrete mathematics. msc | Mathematics education – Founda- tions of mathematics – Logic. msc Classification: LCC QA248 .B9657 2018 | DDC 511/.1–dc23 LC record available at https://lccn.loc.gov/2018023584

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Preface vii What Is Discrete Mathematics? vii Goals of the Book vii Features of the Book viii Course Outline ix Acknowledgments xi

1 Convince Me! 1 1.1 Opening Problems 2 1.2 Solutions 4

2 Mini- 9 2.1 Introduction 9 2.2 Divisibility of 14 2.3 Matrices 22

3 Logic and Sets 31 3.1 Propositions 31 3.2 Sets 33 3.3 Logical Operators and Truth Tables 38 3.4 Operations on Sets 44 3.5 Truth Values of Compound Propositions 51 3.6 Set Identities 54 3.7 Infinite Sets and Paradoxes 58

4 Logic and Proof 67 4.1 Logical Equivalences 67 4.2 Predicates 73 4.3 Nested Quantifiers 77 4.4 Rules of 83 4.5 Methods of Proof 91

5 Relations and Functions 101 5.1 Relations 101 5.2 Properties of Relations on a Set 106 5.3 Functions 113 5.4 123

iii iv Contents

6 Induction 133 6.1 Inductive and Deductive Thinking 133 6.2 Well-Ordering Principle 135 6.3 Method of 137 6.4 Strong Induction 147 6.5 Proof of the Division 151

7 Number Theory 155 7.1 Primes 155 7.2 The Euclidean 159 7.3 Linear Diophantine 165 7.4 Congruences 169 7.5 Applications 176 7.6 Additional Problems 178

8 Counting 183 8.1 What Is Counting? 183 8.2 Counting Techniques 184 8.3 and 193 8.4 The Binomial Theorem 203 8.5 Additional Problems 207

9 Graph Theory 211 9.1 The Language of Graphs 212 9.2 Traversing Edges and Visiting Vertices 221 9.3 Colorings 230 9.4 Trees 238 9.5 Proofs of Euler’s and Ore’s 249

10 Invariants and Monovariants 253 10.1 Invariants 253 10.2 Monovariants 259

11 Topics in Counting 267 11.1 Inclusion-Exclusion 267 11.2 The 274 11.3 Multinomial Coefficients 278 11.4 Combinatorial Identities 282 11.5 Occupancy Problems 286

12 Topics in Number Theory 299 12.1 More on Primes 299 12.2 Integers in Other Bases 303 12.3 More on Congruences 311 12.4 Nonlinear Diophantine Equations 318 12.5 : Rabin’s Method 319 Contents v

13 Topics in Graph Theory 327 13.1 Planar Graphs 327 13.2 Chromatic 331 13.3 Spanning 337 13.4 and Circuit Algorithms 342

Hints 355

List of Names 381

Bibliography 383

Index 385

Preface

What Is Discrete Mathematics? The word discrete in mathematics is in contrast to the word continuous. For example, the set of integers is discrete, while the set of real is continuous. Thus, dis- crete mathematics describes a collection of branches of mathematics with the common characteristic that they focus on the study of things consisting of separate, irreducible, often finite parts. Although largely neglected in typical precollege mathematics cur- ricula, discrete mathematics is essential for developing logic and problem-solving abil- ities. Questions located within the realm of discrete mathematics naturally invite cre- ativity and innovative thinking that go beyond formulas. Furthermore, the cultivation of logical thinking forms a necessary foundation for proof-writing. For these reasons, discrete mathematics is critical for undergraduate study of both mathematics and com- puter science.

Goals of the Book Simply stated, the goal of Journey into Discrete Mathematics is to nurture the develop- ment of skills needed to learn and do mathematics. These skills include the ability to read, write, and appreciate a good , as well as a basic fluency with core mathematical topics such as sets, relations and functions, graph theory, and num- ber theory. The content and the corresponding requisite mathematical thinking are appropriate for students in and other problem-solving disciplines, but the content presentation and the nature of the problem sets reinforce the primary goal of training . Throughout the book, we emphasize the and the essentials of proof-writing, and we underscore that the process is very important in mathematics. Entry-level discrete mathematics serves as an excellent gateway to upper-level math- ematics by priming students’ minds for upper-level concepts. Journey into Discrete Mathematics is designed for use in the first noncalculus course of a mathematics ma- jor, employing a writing style that models a high of mathematical accuracy while maintaining accessibility for early college students. For example, the treatment of inclusion-exclusion provides both informal and technically precise explanations. Ul- timately, the goal behind this approach is communication: we want to model and teach students to communicate both accurately and clearly.

vii viii Preface

Journey into Discrete Mathematics utilizes problems and examples to lay the foun- dation for concepts to be encountered in future mathematics courses. For example, the chapter on relations and functions introduces students to definitions such as one-to- one and onto; several problems in Chapter 4 guide students through definitions of con- tinuity using nested quantifiers; the treatment of greatest common foreshadows finding the GCD of functions; the binomial and multinomial theorems are presented as tools for combinatorial counting; and Euler’s totient and Fer- mat’s Little Theorem are important number-theoretic concepts that students will see again in an abstract course. The homework questions are divided into sections according to difficulty, spanning the gamut from routine to quite challenging. Thefirst section generally includes exercises that are more routine or computational, meant to give students a chance to practice given techniques, while the latter sections generally consist of problems that require creativity, synthesis of multiple concepts, or proofs. This book takes the time to describe the origins of important discrete math top- ics as well as the connections between concepts. The treatment of matrices references ’s first use of matrices; the introduction of Fibonacci numbers is placed within historical context; the work on inductive thinking and proof by induction ex- hibits care for making connections with deductive thinking, the Well-Ordering Princi- ple, and other mathematical concepts. Inspirational quotes throughout the book and the incorporation of the first names of mathematicians in examples and exercises (with a corresponding summary list providing a brief biography for each one mentioned) con- tribute to familiarizing students with the names of key figures within discrete mathe- matics.

Features of the Book Convince Me Chapter. This opening chapter contains a selection of interesting, non- standard problems of varying degrees of difficulty. Readers are invited to think cre- atively and argue persuasively as they work to find solutions. This process cultivates an understanding of the importance of making a good mathematical argument, while setting the tone for the problem-solving nature of the book. Moreover, many of the solutions in this chapter foreshadow the mathematical techniques and theorems that will be encountered later in the book. Hook Problems. In the manner of the Convince Me problems, each chapter begins with an intriguing and challenging problem intended to capture the reader’s interest. Each hook problem can be solved using techniques to be developed in the chapter and usually reappears later in the chapter, either as an example or as a homework problem. Presentation of Logic. Chapters 3 and 4 of the book combine the topics of sets, logic, and proof-writing in a distinctive way. This approach helps to highlight the high level of congruity between concepts such as DeMorgan’s Laws for sets and logic, member- ship tables and truth tables, logical operators and set operators. The chapter on logic and proof-writing appears early in the book to help students the gap between intuitive thinking and the formal presentation of an argument, both of which are nec- essary in mathematics. First Thoughts and Further Thoughts. Solutions to many examples in the book are preceded by “First Thoughts”, describing the initial thought process that one might engage in when first considering a new problem. This is intended to be both helpful Course Outline ix and reassuring to students who might be intimidated by seeing final polished proofs and assuming that “real” mathematicians can produce these immediately, without in- termediary struggles or failed attempts. First Thoughts help train students in the ways that mathematicians actually operate. Similarly, “Further Thoughts” often follow a solution in order to provide additional insight about, alternative approaches to, or ex- tensions of the given solution; once again, the goal is to cultivate a spirit of doing math- ematics. Advanced Topic Chapters. Several core topics (counting, number theory, and graph theory) are addressed twice in this book: first in an introductory chapter covering stan- dard content and later in a chapter with extended optional (often, but not always, more advanced) material. This provides instructors with flexibility to customize the course, depending on their particular goals, or expand beyond a typical first course in discrete mathematics.

Course Outline This book is designed to be used as a stand-alone text for a three-credit or four-credit discrete mathematics course for average to above average math majors who are learn- ing to write proofs; however, since there is more material in this book than can be covered in a single semester, instructors will have to make some choices. For students who have already had an introduction-to-proofs course, select portions of the first six chapters of the book can be covered rather quickly, and the last half of the book can serve as a main text for a junior-level course in combinatorics. If this book is supple- mented with a few extra topics (such as probability, solving recurrence relations, or finite-state machines), then there is enough material for a two-semester in discrete mathematics. With that goal in mind, we suggest one design for such a three-credit course. The second column in Table 0.1 on page x lists the core sections we believe should be cov- ered. We estimate that the core sections can be covered in about thirty-four fifty-minute lectures. The remaining class periods could be used for review days, testing days, and optional sections from the third column. The first column of the table lists sections and chapters containing material that is essential for students to know before cover- ing corresponding sections of the middle two columns. The middle columns contain material that is used in sections and chapters listed in the fourth column, though they may not be absolute prerequisites. For example, although matrices (first addressed in Section 2.3) also appear in Chapter 9 (Graph Theory), one need not study Section 2.3 in order to be able to understand the essential components of Chapter 9. x Preface

Table 0.1. Section priorities and interdependencies

Prerequisite sections and Core Optional → Material used in chapters for → section section 1.1–1.2 2.1 2.2 Chapters 4, 5, 6, 7, 12 2.3 5.1, 5.2, Chapter 9 3.1–3.6 Chapter 4 3.7 2.2, 3.1–3.6 4.1–4.2 proofs throughout book 4.2 4.3 2.2, 3.1–3.6 4.4–4.5 proofs throughout book 2.2 5.1 5.2, 5.3, 9.1 2.2 5.2 7.4, 9.1 2.2 5.3 5.4 6.1, 6.2, Chapter 11 6.1, 6.2 4.2, 5.3 6.3–6.4 proofs throughout later chapters 6.5 2.2, 6.3 7.1 12.1 2.2, 6.3 7.2 Chapter 12 2.2 7.3 12.4 2.2, 6.3 7.4 2.2 7.5–7.6 12.4 8.1–8.3 Chapters 9 and 11, 13.2 8.4 11.3, 11.4 8.5 8.3 9.1 Chapter 10, 13.4 9.2 9.3 13.1, 13.2 9.4 13.1, 13.2 6.4 9.5 Chapter 4 10.1–10.2 3.2, 3.4, 3.6, 8.3 11.1 2.2 11.2 11.3 11.5 Chapter 8 11.4 Chapter 8, 11.3 11.5 7.1 12.1 2.2 12.2 7.4 12.3 7.3, 12.3 12.4 7.4, 12.3 12.5 9.1, 9.4 13.1 6.4, 9.1, 9.3, 9.4 13.2 9.1, 9.4 13.3 9.1, 9.4 13.4 Acknowledgments xi Acknowledgments Many years of effort went into the conception, writing, and publication of this book.We thank our ever-supportive families, including spouses Barbara, Sherwyn, and Beth, for their encouragement and understanding through the entire endeavor. We thank our students for bearing with us through early rough drafts and for pointing out numerous errors. We give special thanks to Daniel Showalter for his from careful read- ing of numerous parts of the book. Our greatest thanks goes to Felix Lazebnik, who was not only an inspiring discrete math teacher and mentor for two of us, but who provided significant vision and content for the book. In particular, Felix contributed heavilyto Section 2.2, Section 6.5, Chapter 7, and Chapter 10. Finally, we thank MAA and AMS, particularly Stephen Kennedy, Stan Seltzer, and Christine Thivierge, for their excellent work in facilitating the publication process.

List of Names

The first names of the following individuals, all of whom have made notable contribu- tions to discrete mathematics, appear in examples and exercises throughout the book. It is virtually impossible to create an exhaustive list, and we make no claims that we have done so. (1) Abraham Fraenkel (1891–1965), a German-born Israeli mathematician, known for his contributions to axiomatic (2) (1815–1852), an English mathematician and writer, believed to have written instructions for the first computer program (3) Béla Bollobás (1943– ), a Hungarian-born British mathematician and a researcher in combinatorics and graph theory who was heavily influenced by Paul Erdős (4) Brahmagupta Bhillamalacarya (589–668), an Indian mathematician, credited with significant advances in arithmetic, algebra, number theory, and (5) Charles Dodgson (1832–1898), an English author and logician, better known by his pen name Lewis Carroll and for authoring Alice’s Adventures in Wonderland (6) (1938– ), an American computer and mathematician, con- sidered by some to be the “father of the ” (7) Endre Szemerédi (1940– ), a Hungarian-American mathematician who has pub- lished in the fields of combinatorics and theoretical computer science (8) Ernst Zermelo (1871–1953), a German logician known for axiomatizing set theory (9) (1946– ), a Chinese-born American mathematician who worked in var- ious aspects of graph theory (10) Felix Lazebnik (1953– ), a USSR-born American mathematician who works in ex- tremal graph theory and combinatorics using algebraic methods, the Ph.D. advisor of author Owen Byer, and the original source of portions of this book (11) (1921–2005), an American mathematician, considered to be one of the founders of modern graph theory (12) Frank Plumpton Ramsey (1903–1930), a British logician and economist, known for proving Ramsey’s Theorem, an important result in (13) Gary Ebert (1947– ), an American combinatorialist who specializes in finite geome- tries, a pioneer of inductive methods in discrete mathematics based on computer models, and the Ph.D. advisor of author Kenneth Wantz (14) (1845–1918), a German mathematician who founded set theory and the first to prove that the real numbers cannot be put in one-to-one correspondence with the natural numbers

381 382 List of Names

(15) Godfrey Hardy (1877–1947), an English number theorist who authored A Mathe- matician’s Apology and a mentor of Ramanujan (16) Gottlob Frege (1848–1925), a German philosopher and mathematician and one of the founders of modern logic (17) Harold N. (Thann) Ward (1936– ), an American mathematician who specializes in algebraic and its connections to designs and and the Ph.D. advisor of author Deirdre Longacher Smeltzer (18) Herb Wilf (1931–2012), an American mathematician who specialized in combina- torics and graph theory and author of generatingfunctionology [21] (19) Jacob Bernoulli (1654–1705), Swiss mathematician after whom Bernoulli numbers are named and author of the probability text Ars Conjectandi (20) Joel Spencer (1946– ), an American combinatorialist, known for his work on prob- abilistic methods in combinatorics (21) Kurt Gödel (1906–1978), an Austrian-born American logician, best known for his two Incompleteness Theorems (22) László Lovász (1948– ), a Hungarian combinatorialist who wrote books on combi- natorial optimization and theory (23) (1823–1891), a German mathematician who worked in number theory and algebra and was critical of Georg Cantor’s work on set theory (24) Leonardo Bonacci, also known as Leonardo of Pisa and Fibonacci (. 1170–c. 1250), an Italian mathematician, considered to be the best mathematician of the Middle Ages, known for popularizing the sequence of Fibonacci numbers in his book Liber Abaci (25) (1960– ), an Israeli mathematician, known for his extensive contribu- tions to combinatorics and theoretical computer science (26) Paul Erdős (1913–1996), a Hungarian-born mathematician considered to be the “father of discrete mathematics”, one of the most prolific mathematicians of the 20th century, and known for traveling the world and collaborating with more than 500 mathematicians on problems in various of mathematics (27) Peter Cameron (1947– ), an Australian mathematician who has written books on algebra, combinatorics, and coding theory (28) Peter Winkler (1946– ), an American mathematician who specializes in combina- torics and probability, known for his love of mathematical puzzles (29) Ron Graham (1935– ), an American mathematician who has worked in a variety of areas of discrete mathematics and a close friend of Paul Erdős (30) (1887–1920), an Indian mathematician who was largely self- taught but made extraordinary contributions to various , most notably number theory (31) Susanna Epp (1943– ), an American mathematician who is the author of the pop- ular text Discrete Mathematics with Applications (32) (1975– ), an American mathematician and 2006 Fields Medal winner who co-discovered the Green-Tao Theorem, which states that there are arbitrarily long arithmetic progressions of prime numbers Bibliography

[1] Martin Aigner and Günter M. Ziegler, Proofs from The Book, 4th ed., Springer-Verlag, Berlin, 2010. MR2569612 [2] George E. Andrews, The theory of partitions, reprint of the 1976 original, Cambridge Mathematical , Cambridge University Press, Cambridge, 1998. MR1634067 [3] George E. Andrews and Kimmo Eriksson, partitions, Cambridge University Press, Cambridge, 2004. MR2122332 [4] Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count, The art of , The Dolciani Mathematical Expositions, vol. 27, Mathematical Association of America, Washington, DC, 2003. MR1997773 [5] J. A. Bondy and U. S. R. Murty, Graph theory, Graduate Texts in Mathematics, vol. 244, Springer, New York, 2008. MR2368647 [6] David M. Burton, The : An introduction, Allyn and Bacon, Inc., Boston, MA, 1985, pages 463 and 515. MR890956 [7] D. Fomin and L. Kurlyandchik. Light at the end of the tunnel. Quantum, March/April 1994. [8] Michael L. Fredman and Robert Endre Tarjan, Fibonacci heaps and their uses in improved network op- timization algorithms, J. Assoc. Comput. Mach. 34 (1987), no. 3, 596–615, DOI 10.1145/28869.28874. MR904195 [9] Dmitri Fomin, Sergey Genkin, and Ilia Itenberg, Mathematical circles (Russian experience), Mathemat- ical World, vol. 7, American Mathematical Society, Providence, RI, 1996. Translated from the Russian and with a foreword by Mark Saul. MR1400887 [10] Andrew Granville and Greg Martin, races, Amer. Math. Monthly 113 (2006), no. 1, 1–33, DOI 10.2307/27641834. MR2202918 [11] P. J. Heawood. -colour theorem. Quarterly Journal of Mathematics, 1890. [12] I. N. Herstein and I. Kaplansky, Matters mathematical, Chelsea Publishing Co., New York, 1978. MR0497399 [13] Agnes M. Herzberg and M. Ram Murty, Sudoku squares and chromatic polynomials, Notices Amer. Math. Soc. 54 (2007), no. 6, 708–717. MR2327972 [14] Y. Ionin and L. Kurlyandchik. Some things never change. Quantum, September/October 1993. [15] Stasys Jukna, Extremal combinatorics, with applications in computer science, Texts in Theoretical Com- puter Science. An EATCS Series, Springer-Verlag, Berlin, 2001. MR1931142 [16] J. E. Littlewood, Distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 1869–1872. [17] Fred S. Roberts, Applied combinatorics, Prentice Hall, Inc., Englewood Cliffs, NJ, 1984. MR735619 [18] Edward Scheinerman, Mathematics: A discrete introduction, 3rd ed., Cengage Learning. [19] Richard P. Stanley, Enumerative combinatorics. Vol. 1, with a foreword by Gian-Carlo Rota; corrected reprint of the 1986 original, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge Univer- sity Press, Cambridge, 1997. MR1442260 [20] H. Weber, Leopold Kronecker (German), Math. Ann. 43 (1893), no. 1, 1–25, DOI 10.1007/BF01446613. MR1510799 [21] Herbert S. Wilf, generatingfunctionology, 3rd ed., A K Peters, Ltd., Wellesley, MA, 2006. MR2172781

383

Index

푘-partition, 289 hypothesis, 61 contradiction, 54 absolute value, 13 contrapositive, 43, 69 acyclic, 238 proof by contrapositive, 93 Addition Rule, 184 converse, 71, 72 adjacency , 217 countable, 60 antecedent, 40 counterexample, 95 argument, 83 cryptosystem associative law, 10, 25, 55, 68 Rabin’s method, 319 of Choice, 63 RSA, 319 , 9, 63, 83 , 221

Bell numbers, 288 De Moivre, Abraham, 146 biconditional, 41 De Morgan’s Law, 55, 67, 68, 76, 270 bijection, 115 De Morgan, Augustus, 67, 137 binomial coefficients, 204 derangement, 272 Binomial Theorem, 204 digraph, 102 Boole, George, 67 , 102 Cantor, Georg, 61, 381 Dijkstra’s Algorithm, 347, 349 , 58 Diophantine , 166, 318 , 45 directed tree, 240 Cayley’s formula, 248 descendent, 240 cell, 109 height, 240 Chinese Remainder Theorem, 317 leaf, 240 chromatic number, 231, 331 level, 240 chromatic polynomial, 334 offspring, 240 circuit, 215, 221 parent, 240 , 214 root, 240 , 10, 25, 91 sibling, 240 , 193 Dirichlet, Peter, 300 combinatorial proof, 282 disjoint sets, 44 commutative law, 10, 25, 55, 68 disjunction, 38 of a set, 44 disjunctive syllogism, 85 Complement Rule, 185 distributive law, 10, 25, 55, 68 , 15, 155 divisible, 14 conclusion, 40, 83 Division Theorem, 17, 21, 151, 304 conditional, 40 divisor, 14 congruence, 108, 311 domain, 102, 120 congruence class, 109, 312 double implication, 41 congruent modulo, 170 dynamic programming algorithm, 347 conjecture, 33, 83, 87 edge, 212 conjunction, 38 , 332 conjunctive simplification, 85 equivalence class, 108 consequent, 40 equivalence , 107, 170

385 386 Index

Erdős, Paul, 382 halting problem, 62 , 160–162, 313, 322 Hamilton cycle, 225 Euler circuit, 222 Hamilton path, 225 Euler path, 222 Handshaking Problem, 214 Euler’s formula, 328 harmonic numbers, 146 Euler’s Theorem, 224, 249 Hilbert, David, 165 Euler, Leonhard, 134, 211, 222, 301, 337 Horner’s Algorithm, 305 hypothesis, 40, 83 , 125 hypothetical syllogism, 85 , 301 Fermat, Pierre, 134, 137, 159, 301 implication, 40, 41, 69 Fibonacci numbers, 126, 130, 143, 144, 148, 179, Inclusion-Exclusion Principle, 185, 268–271 283, 285 inductive hypothesis, 138 Fibonacci, Leonardo de Pisa, 126, 311, 382 injection, 114 , 235, 334 integers, 9 fractal, 131 even, 14, 91 Fraenkel, Abraham, 63, 381 odd, 14, 91 function, 113 , 255 composition, 118 inverse, 71, 72, 117 identity, 119 inversion, 256 image, 113 invertible modulo 푛, 312 inverse, 117 one-to-one, 114 푘-colorable, 231 onto, 115 푘-coloring, 295 Fundamental Theorem of Counting, 186 Koch snowflake, 130 Königsberg Bridge Problem, 222 Gauss, Carl, 299 Kronecker, Leopold, 10, 382 geometric , 129 Kruskal’s Algorithm, 244, 339 Goldbach Conjecture, 33, 302 Kuratowski’s Theorem, 329 Goldbach, Christian, 134 graph, 102, 212 least common multiple, 169 bipartite, 233 least residue, 312 complement, 213 linear combination, 162 complete bipartite, 219 linear congruences, 312 , 214 logical equivalence, 67 , 215 logical fallacies, 88 conflict graph, 230 Mathematical Induction, 133, 137 connected, 215 strong induction, 147 edge, 215 membership table, 45 degree sequence, 264 Mersenne primes, 302 Eulerian, 222 minimum Hamilton cycle, 227 forest, 238 minimum , 337 graphic, 264 mod, 18, 107, 170 homeomorphic, 329 modus ponens, 85, 87 푘-partite, 235, 264, 335 modus tollens, 85, 87 labeled, 215 monovariant, 261 , 212 multinomial coefficient, 205, 278, 280 null, 214 Multinomial Theorem, 279 path, 221 Multiplication Rule, 186 planar, 234 planar, , 327 natural numbers, 9 regular, 213 Nearest Neighbor Algorithm, 344, 349 simple, 212 negation, 38 subdivision, 329 Nim, 308 unlabeled, 215 Nim sum, 309 weighted, 227 , 35 , 236 , 160 occupancy problems, 290, 292 one-to-one correspondence, 115 Index 387

Ore’s Theorem, 226, 250, 264 reflexive, 106 symmetric, 106 palindrome, 192 transitive, 107 parity, 255 relatively prime, 160 partial ordering, 107 remainder, 17 partition, 109 rules of inference, 83, 85 number, 290 conjunctive simplification, 85 of a set, 109, 287, 288 disjunctive syllogism, 85 of an integer, 289 hypothetical syllogism, 85 Pascal’s Triangle, 204 modus ponens, 85, 87 path, 214 modus tollens, 85, 87 simple, 215, 221 Russell’s Paradox, 62 , 193 Russell, Bertrand, 62 , 329 Pick’s Theorem, 150 sequence, 123 Pigeonhole Principle, 274 arithmetic, 126 extension, 276 common difference, 126 generalization, 276 geometric, 127 , 327, 328 common ratio, 127 power set, 45 recursive, 125 predicate, 73 set, 33 Prim’s Algorithm, 341, 349 cardinality, 34 prime number, 15, 155, 156, 299 complement, 44 Prime Number Theorem, 301 countably finite, 60 proof countably infinite, 60 by contradiction, 87, 91, 92, 157 difference, 45 by contrapositive, 91 elements, 34 constructive, 95 empty (or null), 35 direct, 91 equal sets, 35 indirect, 91, 92 equality, 35 proper coloring, 231 equivalent, 58 proposition, 31 intersection, 44 compound, 33 set-builder notation, 34 simple, 33 , 35 propositional form, 73 , 44 universal, 44 quadratic residue, 319 set-builder notation, 74 , 74 shift cipher, 177 existential, 74 shortest-path algorithm, 346 nested, 77 Sierpiński triangle, 131 universal, 74 simple path, 215 quotient, 14, 17 Sorted Edge Algorithm, 345, 349 Rabin’s method, 319 spanning tree, 242, 337 range, 102 Kruskal’s Algorithm, 339 rational numbers, 10 minimum, 243 rearrangement, 197 Prim’s Algorithm, 341 regular , 329 , 31 relation, 101 simple, 33 antisymmetric, 107 Stirling number of the second kind, 287 asymmetric, 107 subgraph, 213 binary, 101 maximally connected, 215 counting them, 208 subset, 35 digraph, 102 proper, 35 equivalence relation, 107 summation notation, 46 graph of, 102 surjection, 115 irreflexive, 106 , 50, 98 matrix, 105 system of congruences, 314 partial ordering, 107 tautology, 54 388 Index theorem, 83 totient function, 179, 315 Tower of Hanoi, 145 Traveling Salesperson Problem, 342 tree, 238 binary, 241 complete , 241 triangular numbers, 128 , 38, 52 , 31, 51 Twin Prime Conjecture, 302 undecidable, 33 unicursive drawing, 223 vacuously true, 96 valid argument, 83 Venn , 47 vertex, 212 adjacent, 212 degree, 212 in-degree, 102 isolated, 212 neighbor, 212 out-degree, 102

Waring Conjecture, 302 Well-Ordering Principle, 135 Wilf, Herbert, 133

Zermelo, Ernst, 63, 381 AMS / MAA TEXTBOOKS VOL 41 Journey into VOL Discrete Mathematics AMS / MAA TEXTBOOKS 41

Owen D. Byer, Deirdre L. Smeltzer, and Kenneth L. Wantz Journey into Discrete Mathematics Journey into Discrete Mathematics is designed for use in a fi rst course in mathematical abstraction for early-career undergraduate mathematics

majors. The important ideas of discrete mathematics are included— Owen D. Byer, Deirdre L. Smeltzer, and Kenneth L. Wantz logic, sets, proof writing, relations, counting, number theory, and graph theory—in a manner that promotes development of a mathematical mindset and prepares students for further study. While the treatment is designed to prepare the student reader for the mathematics major, the book remains attractive and appealing to students of computer science and other problem-solving disciplines. The exposition is exquisite and engaging and features detailed descrip- tions of the thought processes that one might follow to attack the problems of mathematics. The problems are appealing and vary widely in depth and diffi culty. Careful design of the book helps the student reader learn to think like a mathematician through the exposition and the prob- lems provided. Several of the core topics, including counting, number theory, and graph theory, are visited twice: once in an introductory manner and then again in a later chapter with more advanced concepts and with a deeper perspective. Owen D. Byer and Deirdre L. Smeltzer are both Professors of Mathematics at Eastern Mennonite University. Kenneth L. Wantz is Professor of Mathematics at Regent University. Collectively the authors have special- ized expertise and research publications ranging widely over discrete mathematics and have over fi fty semesters of combined experience in teaching this subject.

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