Handbook of Mathematical Induction

Handbook of Mathematical Induction : Theory and Applications

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Description

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn's lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process.show more

Product details

  • Hardback | 921 pages
  • 185.42 x 254 x 50.8mm | 1,723.64g
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • New.
  • N/A; N/A CR Book; 9 Tables, black and white; 38 Illustrations, black and white
  • 1420093649
  • 9781420093643
  • 1,847,729

Review quote

... a treasure trove for anyone who is ... interested in mathematics as a hobby, or as the target of proof automation or assistance. It could also be the basis for a crosscutting course in mathematics, based on seeing how one can apply a single proof technique across the field.- Simon Thompson in Computing News, May 2011 Gunderson started out collecting some induction problems for discrete math students and then couldn't stop himself, thereafter assembling more than 750 of the addictive things for this handbook and supplementing them with a grounding in theory and discussion of applications. He offers 500-plus complete solutions, and many of the other problems come with hints or references; unlike other treatments, this handbook treats the subject seriously and is not just a `collection of recipes'. It's a book that will work well with most math or computing science courses, on a subject that pertains to graph theory, point set topology, elementary number theory, linear algebra, analysis, probability theory, geometry, group theory, and game theory, among many other topics.-SciTech Book News, February 2011 ... a unique work ... the ostensibly narrow subject of mathematical induction is carefully and systematically expounded, from its more elementary aspects to some quite sophisticated uses of the technique. This is done with a (very proper!) emphasis on solving problems by means of some form of induction or other ... any of us who regularly teach the undergraduate course aimed at introducing mathematics majors to methods of proof quite simply need to own this book. ... In this boot camp course, it is imperative that problems should be abundant, both in supply and variety, and should be capable of careful dissection. Gunderson hit[s] the mark on both counts ... Gunderson's discussions are evocative and thorough and can be appreciated by mathematicians of all sorts ... [he] develop[s] the requisite surrounding material with great care, considerably enhancing the value of his book as a supplementary text for a huge number of courses, both at an undergraduate and graduate level ... a very welcome addition to the literature ...-MAA Reviews, December 2010show more

About David S. Gunderson

David S. Gunderson is a professor and chair of the Department of Mathematics at the University of Manitoba in Winnipeg, Canada. He earned his Ph.D. in pure mathematics from Emory University. His research interests include Ramsey theory, extremal graph theory, combinatorial geometry, combinatorial number theory, and lattice theory.show more

Table of contents

THEORYWhat 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 FoundationsNotation Axioms Peano's axioms Principle of mathematical induction Properties of natural numbers Well-ordered sets Well-founded sets Variants of Finite Mathematical Induction The first principle Strong mathematical induction Downward induction Alternative forms of mathematical induction Double induction Fermat's method of infinite descent Structural induction Inductive Techniques Applied to the Infinite More on well-ordered sets Transfinite induction Cardinals Ordinals Axiom of choice and its equivalent forms Paradoxes and Sophisms from InductionTrouble with the language? Fuzzy definitions Missed a case? More deceit? Empirical InductionIntroduction 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 â 1Goodstein sequences How to Prove by InductionTips on proving by induction Proving more can be easier Proving limits by induction Which kind of induction is preferable? The Written MI ProofA template Improving the flow Using notation and abbreviations APPLICATIONS AND EXERCISESIdentitiesArithmetic 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 Gaussian coefficients Trigonometry identities Miscellaneous identities Inequalities Number TheoryPrimes Congruences Divisibility Numbers expressible as sums Egyptian fractions Farey fractions Continued fractions SequencesDifference sequences Fibonacci numbers Lucas numbers Harmonic numbers Catalan numbers Schroder numbers Eulerian numbers Euler numbers Stirling numbers of the second kind SetsProperties of sets Posets and lattices Topology Ultrafilters Logic and LanguageSentential logic Equational logic Well-formed formulae Language GraphsGraph theory basics Trees and forests Minimum spanning trees Connectivity, walks Matchings Stable marriages Graph coloring Planar graphs Extremal graph theory Digraphs and tournaments Geometric graphs Recursion and AlgorithmsRecursively defined operations Recursively defined sets Recursively defined sequences Loop invariants and algorithms Data structures Complexity Games and RecreationsIntroduction to game theory Tree games Tiling with dominoes and trominoes Dirty faces, cheating wives, muddy children, and colored hats Detecting a counterfeit coin More recreations Relations and FunctionsBinary relations Functions Calculus Polynomials Primitive recursive functions Ackermann's function Linear and Abstract AlgebraMatrices and linear equations Groups and permutations Rings Fields Vector spaces GeometryConvexity 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 StatisticsProbability basics Basic probability exercises Branching processes The ballot problem and the hitting game Pascal's game Local lemma SOLUTIONS AND HINTS TO EXERCISESFoundations Empirical Induction Identities Inequalities Number Theory Sequences SetsLogic and Language Graphs Recursion and Algorithms Games and Recreation Relations and FunctionsLinear and Abstract Algebra Geometry Ramsey Theory Probability and Statistics APPENDICESZFC Axiom SystemInducing You to Laugh?The Greek Alphabet References Indexshow more

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