A Gentle Introduction to the Art of Mathematics Version 3.1 Joseph Fields Southern Connecticut State University ii Copyright c 2013 Joseph E. Fields. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no In- variant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled \GNU Free Documentation License". The latest version of this book is available (without charge) in portable document format at http://www.southernct.edu/~fields/ iii Acknowledgments This is version 3.1 of A Gentle Introduction to the Art of Mathe- matics. Earlier versions were used and classroom tested by several colleagues: Robert Vaden-Goad, John Kavanagh, Ross Gingrich, Aaron Clark. I thank you all. A particular debt of gratitude is owed to Len Brin whose keen eyes caught a number of errors and inconsistencies, and who contributed many new exercises. Thanks, Len. A word about options There are currently 4 optional versions of this text. The different options are distinguished with flags following the version number. Thus you may see a plain \version 3.1" (no flags) or any of the fol- lowing: \version 3.1S", \version 3.1N", or \version 3.1SN". The S flag is for those who prefer the symbol ∼ to represent logical nega- tion. (The default is to use the symbol : for logical negation.) The N flag distinguishes a version in which the convention that 1 is the smallest natural number is followed throughout. The absence of the N flag indicates the original version of the text in which the naturals do not contain 0 in chapters 1 through 4 and (after the Peano axioms are introduced) that convention is changed to the more modern rule. iv Contents 1 Introduction and notation1 1.1 Basic sets.............................1 1.2 Definitions: Prime numbers................... 12 1.3 More scary notation....................... 20 1.4 Definitions of elementary number theory............ 23 1.4.1 Even and odd....................... 23 1.4.2 Decimal and base-n notation............... 23 1.4.3 Divisibility......................... 25 1.4.4 Floor and ceiling..................... 26 1.4.5 Div and mod....................... 27 1.4.6 Binomial coefficients................... 29 1.5 Some algorithms......................... 36 1.6 Rational and irrational numbers................. 45 1.7 Relations.............................. 50 2 Logic and quantifiers 55 2.1 Predicates and Logical Connectives............... 55 2.2 Implication............................ 68 2.3 Logical equivalences........................ 74 2.4 Two-column proofs........................ 88 2.5 Quantified statements...................... 92 2.6 Deductive reasoning and Argument forms............ 101 v vi CONTENTS 2.7 Validity of arguments and common errors............ 110 3 Proof techniques I 119 3.1 Direct proofs of universal statements.............. 119 3.2 More direct proofs........................ 132 3.3 Contradiction and contraposition................ 137 3.4 Disproofs............................. 143 3.5 By cases and By exhaustion................... 148 3.6 Existential statements...................... 157 4 Sets 165 4.1 Basic notions of set theory.................... 165 4.2 Containment........................... 172 4.3 Set operations........................... 177 4.4 Venn diagrams.......................... 188 4.5 Russell's Paradox......................... 197 5 Proof techniques II | Induction 201 5.1 The principle of mathematical induction............ 201 5.2 Formulas for sums and products................. 211 5.3 Other proofs using PMI..................... 222 5.4 The strong form of mathematical induction........... 230 6 Relations and functions 233 6.1 Relations.............................. 233 6.2 Properties of relations...................... 243 6.3 Equivalence relations....................... 251 6.4 Ordering relations......................... 260 6.5 Functions............................. 270 6.6 Special functions......................... 283 CONTENTS vii 7 Proof techniques III | Combinatorics 293 7.1 Counting.............................. 293 7.2 Parity and Counting arguments................. 309 7.3 The pigeonhole principle..................... 323 7.4 The algebra of combinations................... 328 8 Cardinality 339 8.1 Equivalent sets.......................... 339 8.2 Examples of set equivalence................... 345 8.3 Cantor's theorem......................... 357 8.4 Dominance............................ 366 8.5 CH and GCH........................... 375 9 Proof techniques IV | Magic 381 9.1 Morley's miracle.......................... 383 9.2 Five steps into the void...................... 390 9.3 Monge's circle theorem...................... 402 References 410 Index 410 viii CONTENTS List of Figures 1.1 The sieve of Eratosthenes..................... 15 1.2 Pascal's triangle.......................... 30 1.3 A small example in pseudocode and as a flowchart....... 38 1.4 The division algorithm in flowchart form............. 40 1.5 The Euclidean algorithm in flowchart form............ 42 2.1 A schematic representation of a transistor............ 60 2.2 Series connections implement and................. 61 2.3 Parallel connections implement or................ 61 2.4 Parenthesizations expressed as digital logic circuits....... 63 2.5 Disjunctive normal form...................... 65 3.1 A four-color map.......................... 149 3.2 Graph pebbling........................... 152 3.3 Graph pebbling move....................... 153 3.4 A Z-module............................. 161 6.1 An example of a relation...................... 234 6.2 An example of the \divides" relation............... 235 6.3 The graph of the \less than" relation............... 239 6.4 The graph of the divisibility relation............... 239 6.5 Some simple Hasse diagrams................... 263 6.6 Hasse diagram for (P(f1; 2; 3g); ⊆)................ 264 ix x LIST OF FIGURES 6.7 Hasse diagram of divisors of 72.................. 265 6.8 The sets related to an arbitrary function............. 271 7.1 Full houses in Yahtzee....................... 296 7.2 K¨onigsberg, Prussia........................ 313 7.3 K¨onigsberg, Prussia as a graph.................. 314 7.4 A desk with pigeonholes...................... 323 8.1 Cantor's snake........................... 349 8.2 Equivalent intervals........................ 352 8.3 An interval is equivalent to a semi-circle............. 353 8.4 Binary representations in the unit interval............ 359 8.5 Setup for proving the C-B-S theorem............... 370 8.6 An A-stopper in the proof of C-B-S................ 372 9.1 The setup for Morley's Miracle.................. 384 9.2 The first Morley triangle...................... 385 9.3 Conway's puzzle proof....................... 388 9.4 Scaling in Conway's puzzle proof................. 388 9.5 An infinite army in the lower half-plane............. 391 9.6 Moving one step into the void is trivial.............. 393 9.7 Moving two steps into the void is more difficult......... 394 9.8 Moving three steps into the void takes 8 men.......... 395 9.9 The taxicab distance to (0; 5)................... 397 9.10 Finding r.............................. 398 9.11 Setup for Monge's circle theorem................. 403 9.12 Example of Monge's circle theorem................ 405 9.13 Four triangles bounded by 6 line segments........... 407 List of Tables 2.1 Converse, inverse and contrapositive............... 71 2.2 Basic logical equivalences..................... 83 2.3 The rules of inference....................... 106 3.1 The definitions of elementary number theory restated...... 124 4.1 Basic set theoretic equalities.................... 180 6.1 Properties of relations....................... 244 xi xii LIST OF TABLES To the student You are at the right place in your mathematical career to be reading this book if you liked Trigonometry and Calculus, were able to solve all the prob- lems, but felt mildly annoyed with the text when it put in these verbose, incomprehensible things called \proofs." Those things probably bugged you because a whole lot of verbiage (not to mention a sprinkling of epsilons and deltas) was wasted on showing that a thing was true, which was obviously true! Your physical intuition is sufficient to convince you that a statement like the Intermediate Value Theorem just has to be true { how can a function move from one value at a to a different value at b without passing through all the values in between? Mathematicians discovered something fundamental hundreds of years be- fore other scientists { physical intuition is worthless in certain extreme sit- uations. Probably you've heard of some of the odd behavior of particles in Quantum Mechanics or General Relativity. Physicists have learned, the hard way, not to trust their intuitions. At least, not until those intuitions have been retrained to fit reality! Go back to your Calculus textbook and look up the Intermediate Value Theorem. You'll probably be surprised to find that it doesn't say anything about all functions, only those that are continuous. So what, you say, aren't most functions continuous? Actually, the number of functions that aren't continuous represents an infinity so huge that it outweighs the infinity of the real numbers! The point of this book is to help you with the transition from doing math xiii xiv TO THE STUDENT at an elementary level (which is concerned mostly with solving problems) to doing math