دانلود کتاب Handbook of Mathematical Induction: Theory and Applications

FeaturesPresents hundreds of classical theorems and proofs that span many areas, including basic equalities and inequalities, combinatorics, linear algebra, calculus, trigonometry, geometry, set theory, game theory, recursion, and algorithmsDerives many forms of mathematical induction, such as infinite descent and the axiom of choice, from basic principlesSupplies all necessary definitions and background, thereby requiring only a very modest amount of mathematical maturity to understand most results and proofsContains more than 750 exercises, with complete solutions to at least 500Includes nearly 600 bibliographic references, numerous cross references, and an extensive index of over 3,000 entriesHandbook 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.Table of ContentsTHEORYWhat 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: diceGauss and sums A variety of applications History of mathematical induction Mathematical induction in modern literatureFoundationsNotation Axioms Peano’s axioms Principle of mathematical induction Properties of natural numbers Well-ordered sets Well-founded setsVariants 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 inductionInductive Techniques Applied to the Infinite More on well-ordered sets Transfinite induction Cardinals Ordinals Axiom of choice and its equivalent formsParadoxes 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 − 1Goodstein sequencesHow 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 abbreviationsAPPLICATIONS 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 identitiesInequalitiesNumber TheoryPrimes Congruences Divisibility Numbers expressible as sums Egyptian fractions Farey fractions Continued fractionsSequencesDifference sequences Fibonacci numbers Lucas numbers Harmonic numbers Catalan numbers Schröder numbers Eulerian numbers Euler numbers Stirling numbers of the second kindSetsProperties of sets Posets and lattices Topology UltrafiltersLogic and LanguageSentential logic Equational logic Well-formed formulae LanguageGraphsGraph theory basics Trees and forests Minimum spanning trees Connectivity, walks Matchings Stable marriages Graph coloring Planar graphs Extremal graph theory Digraphs and tournaments Geometric graphsRecursion and AlgorithmsRecursively defined operations Recursively defined sets Recursively defined sequences Loop invariants and algorithms Data structures ComplexityGames 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 recreationsRelations and FunctionsBinary relations Functions Calculus Polynomials Primitive recursive functions Ackermann’s functionLinear and Abstract AlgebraMatrices and linear equations Groups and permutations Rings Fields Vector spacesGeometryConvexity Polygons Lines, planes, regions, and polyhedra Finite geometriesRamsey Theory The Ramsey arrow Basic Ramsey theorems Parameter words and combinatorial spaces Shelah bound High chromatic number and large girthProbability and StatisticsProbability basics Basic probability exercises Branching processes The ballot problem and the hitting game Pascal’s game Local lemmaSOLUTIONS 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 StatisticsAPPENDICESZFC Axiom SystemInducing You to Laugh?The Greek AlphabetReferencesIndex