Series Editor KENNETH H. ROSEN INDUCTION THEORY AND APPLICATIONS. of Manitoba. University. Winnipeg, Canada. CRC Press. Taylor StFrancis Group

Transcription

1 DISCIIETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN HANDBOOK OF MATHEMATICAL INDUCTION THEORY AND APPLICATIONS David S. Gunderson University of Manitoba Winnipeg, Canada CRC Press Taylor StFrancis Group Boca Raton London NewYork CRC Press is an imprint of the Taylor St Francis an Group, informa business A CHAPMAN & HALL BOOK

2 Contents Foreword xvii Preface xix About the author xxv I Theory 1 What is mathematical induction? Introduction An informal introduction to mathematical induction Ingredients of a proof by mathematical induction Two other ways to think of mathematical induction A simple example: Dice Gauss and sums A variety of applications History of mathematical induction Mathematical induction in modern literature 15 2 Foundations Notation Axioms Peano's axioms Principle of mathematical induction Properties of natural numbers Well-ordered sets Well-founded sets 33 3 Variants of finite mathematical induction The first principle Strong mathematical induction Downward induction 38 vii

3 1 viii Contents 3.4 Alternative forms of mathematical induction Double induction Format's method of infinite descent Structural induction 48 4 Inductive techniques applied to the infinite More on well-ordered sets Transrinite induction Cardinals Ordinals 5'5 4.5 Axiom of choice and its equivalent forms 57 5 Paradoxes and sophisms from induction Trouble with the language? Richard's paradox Paradox of the unexpected exam Fuzzy definitions No crowds allowed Nobody is rich Everyone is bald Missed a case? All is for naught All horses are the same color Non-parallel lines go through one point More deceit? A new formula for triangular numbers All positive integers are equal Four weighings suffice 74 6 Empirical induction Introduction Guess the pattern? A pattern in primes? A sequence of integers? Sequences with only primes? Divisibility Never a square? Goldbach's conjecture Cutting the cake Sums of hex numbers Factoring xn Goodstein sequences 86

4 Contents ix 7 How to prove by induction Tips on proving by induction Proving more can be easier Proving limits by induction Which kind of induction is preferable? When is induction needed? Which kind of induction to use? The written MI proof A template Improving the flow Using other results in a proof Clearly, it's trivial! Pronouns Footnotes We, let's, our, will, now, must Using notation and abbreviations 116 II Applications and exercises 9 Identities Arithmetic progressions Sums of finite geometric series and related series Power sums, sums of a single power Products and sums of products Sums or products of fractions Identities with binomial coefficients Abel identities Bernoulli numbers Faulhaber's formula for power sums Gaussian coefficients Trigonometry identities Miscellaneous identities Inequalities Number theory Primes Congruences Divisibility Numbers expressible as sums Egyptian fractions 176

5 x Contents 11.6 Farcy fractions Continued fractions Finite continued fractions Infinite continued fractions Sequences Difference sequences Fibonacci numbers Lucas numbers Harmonic numbers Catalan numbers Introduction Catalan numbers denned by a formula C as a number of ways to compute a product The definitions are equivalent Some occurrences of Catalan numbers Schroder numbers Eulerian numbers Ascents, descents, rises, falls Definitions for Eulerian numbers Eulerian number exercises Euler numbers Stirling numbers of the second kind Sets Properties of sets Posets and lattices Topology Ultrafilters Logic and language Sentential logic Equational logic Well-formed formulae 235 J 4.4 Language Graphs Graph theory basics Trees and forests Minimum spanning trees Connectivity, walks Matching* Stable marriages 250

6 Contents xi 15.7 Graph coloring Planar graphs Extremal graph theory Digraphs and tournaments Geometric graphs Recursion and algorithms Recursively defined operations Recursively defined sets Recursively defined sequences Linear homogeneous recurrences of order Method of characteristic roots Applying the method of characteristic roots Linear homogeneous recurrences of higher order Non-homogeneous recurrences Finding recurrences Non-linear recurrence Towers of Hanoi Loop invariants and algorithms Data structures Gray codes The hypercube Red-black trees Complexity Landau notation The master theorem Closest pair of points Games and recreations Introduction to game theory Tree games Definitions and terminology The game of NIM Chess Tiling with dominoes and trorninoes Dirty faces, cheating wives, muddy children, and colored hats A parlor game with sooty fingers Unfaithful wives The muddy children puzzle Colored hats More related puzzles and references Detecting a counterfeit coin More recreations 304

7 xii Contents Pennies in boxes Josephus problem The gossip problem Cars on a circular track Relations and functions Binary relations Functions Calculus Derivatives Differential equations Integration Polynomials Primitive recursive functions Ackermann's function Linear and abstract algebra Matrices and linear equations Groups and permutations Semigroups and groups Permutations Rings Fields Vector spaces Geometry Convexity Polygons Lines, planes, regions, and polyhedra Finite geometries Ramsey theory The Ramsey arrow Basic Ramsey theorems Parameter words and combinatorial spaces Shelah bound High chromatic number and large girth Probability and statistics Probability basics Probability spaces Independence and random variables Expected value and conditional probability 390

8 Contents xiii Conditional expectation Basic probability exercises Branching processes The ballot problem and the hitting game Pascal's game 39G 22.6 Local Lemma 397 III Solutions and hints to exercises 23 Solutions: Foundations Solutions: Properties of natural numbers Solutions: Well-ordered sets Solutions: Fermat's method of infinite descent Solutions: Inductive techniques applied to the infinite Solutions: More on well-ordered sets Solutions: Axiom of choice and equivalent forms Solutions: Paradoxes and sophisms Solutions: Trouble with the language? Solutions: Missed a case? 41G 25.3 Solutions: More deceit? 41C 26 Solutions: Empirical induction Solutions: Introduction Solutions: A sequence of integers? Solutions: Sequences with only primes? Solutions: Divisibility Solutions: Never a square? Solutions: Goldbach's conjecture Solutions: Cutting the cake Solutions: Sums of hex numbers Solutions: Identities Solutions: Arithmetic progressions Solutions: Sums with binomial coefficients Solutions: Trigonometry Solutions: Miscellaneous identities Solutions: Inequalities 515

9 xiv Contents 29 Solutions: Number theory Solutions: Primes Solutions: Congruences Solutions: Divisibility Solutions: Expressible as sums Solutions: Egyptian fractions Solutions: Farey fractions Solutions: Continued fractions Solutions: Sequences Solutions: Difference sequences Solutions: Fibonacci numbers Solutions: Lucas numbers Solutions: Harmonic numbers Solutions: Catalan numbers Solutions: Eulerian numbers Solutions: Euler numbers Solutions: Stirling numbers Solutions: Sets Solutions: Properties of sets Solutions: Posets and lattices Solutions: Countable Zorn's lemma for measurable sets Solutions: Topology Solutions: Ultrafilters Solutions: Logic and language Solutions: Sentential logic Solutions: Well-formed formulae Solutions: Graphs Solutions: Graph theory basics Solutions: Trees and forests Solutions: Connectivity, walks Solutions: Match ings Solutions: Stable matchings Solutions: Graph coloring Solutions: Planar graphs Solutions: Extremal graph theory Solutions: Digraphs and tournaments Solutions: Geometric graphs 700

10 Contents xv 34 Solutions: Recursion and algorithms Solutions: Recursively defined sets Solutions: Recursively defined sequences Solutions: Linear homogeneous recurrences of order Solutions: Applying the method of characteristic roots Solutions: Linear homogeneous recurrences of higher order Solutions: Non-homogeneous recurrences Solutions: Non-linear recurrences Solutions: Towers of Hanoi Solutions: Data structures Solutions: Complexity Solutions: Games and recreation Solutions: Tree games Solutions: The game of NIM Solutions: Chess Solutions: Dominoes and trominoes Solutions: Muddy children Solutions: Colored hats Solutions: Detecting counterfeit coin Solutions: Relations and functions Solutions: Binary relations Solutions: Functions Solutions: Linear and abstract algebra Solutions: Linear algebra Solutions: Groups and permutations Solutions: Rings Solutions: Fields Solutions: Vector spaces Solutions: Geometry Solutions: Convexity Solutions: Polygons Solutions: Lines, planes, regions, and polyhedra Solutions: Finite geometries Solutions: Ramsey theory Solutions: Probability and statistics 803

11 xvi Contents IV Appendices Appendix A: ZFC axiom system 815 Appendix B: Inducing you to laugh? 817 Appendix C: The Greek alphabet 821 References 823 Name index 865 Subject index 877