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E-raamat: Handbook of Mathematical Induction: Theory and Applications

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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, Zorns 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.

Arvustused

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. Its 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 Gundersons 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 2010

Foreword xvii
Preface xix
About the author xxv
I Theory
1 What is mathematical induction?
1(18)
1.1 Introduction
1(1)
1.2 An informal introduction to mathematical induction
2(1)
1.3 Ingredients of a proof by mathematical induction
3(1)
1.4 Two other ways to think of mathematical induction
4(1)
1.5 A simple example: Dice
5(1)
1.6 Gauss and sums
6(3)
1.7 A variety of applications
9(2)
1.8 History of mathematical induction
11(4)
1.9 Mathematical induction in modern literature
15(4)
2 Foundations
19(16)
2.1 Notation
19(1)
2.2 Axioms
20(2)
2.3 Peano's axioms
22(1)
2.4 Principle of mathematical induction
23(1)
2.5 Properties of natural numbers
24(6)
2.6 Well-ordered sets
30(3)
2.7 Well-founded sets
33(2)
3 Variants of finite mathematical induction
35(16)
3.1 The first principle
35(1)
3.2 Strong mathematical induction
36(2)
3.3 Downward induction
38(4)
3.4 Alternative forms of mathematical induction
42(1)
3.5 Double induction
43(3)
3.6 Fermat's method of infinite descent
46(2)
3.7 Structural induction
48(3)
4 Inductive techniques applied to the infinite
51(18)
4.1 More on well-ordered sets
51(2)
4.2 Transfinite induction
53(1)
4.3 Cardinals
54(1)
4.4 Ordinals
55(2)
4.5 Axiom of choice and its equivalent forms
57(12)
5 Paradoxes and sophisms from induction
69(8)
5.1 Trouble with the language?
70(1)
5.1.1 Richard's paradox
70(1)
5.1.2 Paradox of the unexpected exam
70(1)
5.2 Fuzzy definitions
71(1)
5.2.1 No crowds allowed
71(1)
5.2.2 Nobody is rich
71(1)
5.2.3 Everyone is bald
71(1)
5.3 Missed a case?
71(2)
5.3.1 All is for naught
72(1)
5.3.2 All horses are the same color
72(1)
5.3.3 Non-parallel lines go through one point
72(1)
5.4 More deceit?
73(4)
5.4.1 A new formula for triangular numbers
73(1)
5.4.2 All positive integers are equal
74(1)
5.4.3 Four weighings suffice
74(3)
6 Empirical induction
77(12)
6.1 Introduction
77(3)
6.2 Guess the pattern?
80(1)
6.3 A pattern in primes?
80(1)
6.4 A sequence of integers?
80(1)
6.5 Sequences with only primes?
81(1)
6.6 Divisibility
82(1)
6.7 Never a square?
83(1)
6.8 Goldbach's conjecture
83(1)
6.9 Cutting the cake
83(1)
6.10 Sums of hex numbers
84(1)
6.11 Factoring xn - 1
85(1)
6.12 Goodstein sequences
86(3)
7 How to prove by induction
89(20)
7.1 Tips on proving by induction
89(5)
7.2 Proving more can be easier
94(3)
7.3 Proving limits by induction
97(6)
7.4 Which kind of induction is preferable?
103(6)
7.4.1 When is induction needed?
103(3)
7.4.2 Which kind of induction to use?
106(3)
8 The written MI proof
109(16)
8.1 A template
110(3)
8.2 Improving the flow
113(3)
8.2.1 Using other results in a proof
113(1)
8.2.2 Clearly, it's trivial!
114(1)
8.2.3 Pronouns
115(1)
8.2.4 Footnotes
115(1)
8.2.5 We, let's, our, will, now, must
115(1)
8.3 Using notation and abbreviations
116(9)
II Applications and exercises
9 Identities
125(28)
9.1 Arithmetic progressions
125(3)
9.2 Sums of finite geometric series and related series
128(1)
9.3 Power sums, sums of a single power
129(2)
9.4 Products and sums of products
131(1)
9.5 Sums or products of fractions
132(2)
9.6 Identities with binomial coefficients
134(10)
9.6.1 Abel identities
141(1)
9.6.2 Bernoulli numbers
141(1)
9.6.3 Faulhaber's formula for power sums
142(2)
9.7 Gaussian coefficients
144(1)
9.8 Trigonometry identities
145(4)
9.9 Miscellaneous identities
149(4)
10 Inequalities
153(8)
11 Number theory
161(26)
11.1 Primes
161(6)
11.2 Congruences
167(3)
11.3 Divisibility
170(6)
11.4 Numbers expressible as sums
176(1)
11.5 Egyptian fractions
176(2)
11.6 Farey fractions
178(1)
11.7 Continued fractions
179(8)
11.7.1 Finite continued fractions
181(3)
11.7.2 Infinite continued fractions
184(3)
12 Sequences
187(30)
12.1 Difference sequences
188(2)
12.2 Fibonacci numbers
190(10)
12.3 Lucas numbers
200(1)
12.4 Harmonic numbers
201(2)
12.5 Catalan numbers
203(5)
12.5.1 Introduction
203(1)
12.5.2 Catalan numbers defined by a formula
203(1)
12.5.3 Cn as a number of ways to compute a product
204(1)
12.5.4 The definitions are equivalent
205(2)
12.5.5 Some occurrences of Catalan numbers
207(1)
12.6 Schroder numbers
208(1)
12.7 Eulerian numbers
209(4)
12.7.1 Ascents, descents, rises, falls
209(1)
12.7.2 Definitions for Eulerian numbers
210(2)
12.7.3 Eulerian number exercises
212(1)
12.8 Euler numbers
213(1)
12.9 Stirling numbers of the second kind
214(3)
13 Sets
217(16)
13.1 Properties of sets
217(6)
13.2 Posets and lattices
223(3)
13.3 Topology
226(3)
13.4 Ultrafilters
229(4)
14 Logic and language
233(6)
14.1 Sentential logic
233(2)
14.2 Equational logic
235(1)
14.3 Well-formed formulae
235(1)
14.4 Language
236(3)
15 Graphs
239(22)
15.1 Graph theory basics
239(3)
15.2 Trees and forests
242(4)
15.3 Minimum spanning trees
246(1)
15.4 Connectivity, walks
247(2)
15.5 Matchings
249(1)
15.6 Stable marriages
250(2)
15.7 Graph coloring
252(1)
15.8 Planar graphs
253(2)
15.9 Extremal graph theory
255(2)
15.10 Digraphs and tournaments
257(1)
15.11 Geometric graphs
258(3)
16 Recursion and algorithms
261(28)
16.1 Recursively defined operations
262(1)
16.2 Recursively defined sets
262(1)
16.3 Recursively defined sequences
263(14)
16.3.1 Linear homogeneous recurrences of order 2
265(1)
16.3.2 Method of characteristic roots
266(3)
16.3.3 Applying the method of characteristic roots
269(1)
16.3.4 Linear homogeneous recurrences of higher order
270(1)
16.3.5 Non-homogeneous recurrences
271(1)
16.3.6 Finding recurrences
272(1)
16.3.7 Non-linear recurrence
273(3)
16.3.8 Towers of Hanoi
276(1)
16.4 Loop invariants and algorithms
277(3)
16.5 Data structures
280(4)
16.5.1 Gray codes
281(1)
16.5.2 The hypercube
281(1)
16.5.3 Red-black trees
282(2)
16.6 Complexity
284(5)
16.6.1 Landau notation
284(1)
16.6.2 The master theorem
285(1)
16.6.3 Closest pair of points
286(3)
17 Games and recreations
289(20)
17.1 Introduction to game theory
289(1)
17.2 Tree games
290(4)
17.2.1 Definitions and terminology
290(2)
17.2.2 The game of NIM
292(1)
17.2.3 Chess
293(1)
17.3 Tiling with dominoes and trominoes
294(1)
17.4 Dirty faces, cheating wives, muddy children, and colored hats
295(5)
17.4.1 A parlor game with sooty fingers
295(1)
17.4.2 Unfaithful wives
296(2)
17.4.3 The muddy children puzzle
298(1)
17.4.4 Colored hats
299(1)
17.4.5 More related puzzles and references
300(1)
17.5 Detecting a counterfeit coin
300(4)
17.6 More recreations
304(5)
17.6.1 Pennies in boxes
304(1)
17.6.2 Josephus problem
304(3)
17.6.3 The gossip problem
307(1)
17.6.4 Cars on a circular track
307(2)
18 Relations and functions
309(16)
18.1 Binary relations
309(1)
18.2 Functions
310(4)
18.3 Calculus
314(6)
18.3.1 Derivatives
314(1)
18.3.2 Differential equations
315(1)
18.3.3 Integration
316(4)
18.4 Polynomials
320(2)
18.5 Primitive recursive functions
322(1)
18.6 Ackermann's function
322(3)
19 Linear and abstract algebra
325(24)
19.1 Matrices and linear equations
325(9)
19.2 Groups and permutations
334(4)
19.2.1 Semigroups and groups
334(1)
19.2.2 Permutations
335(3)
19.3 Rings
338(1)
19.4 Fields
338(3)
19.5 Vector spaces
341(8)
20 Geometry
349(16)
20.1 Convexity
350(4)
20.2 Polygons
354(4)
20.3 Lines, planes, regions, and polyhedra
358(5)
20.4 Finite geometries
363(2)
21 Ramsey theory
365(22)
21.1 The Ramsey arrow
366(1)
21.2 Basic Ramsey theorems
367(6)
21.3 Parameter words and combinatorial spaces
373(5)
21.4 Shelah bound
378(5)
21.5 High chromatic number and large girth
383(4)
22 Probability and statistics
387(18)
22.1 Probability basics
387(5)
22.1.1 Probability spaces
388(1)
22.1.2 Independence and random variables
389(1)
22.1.3 Expected value and conditional probability
390(1)
22.1.4 Conditional expectation
391(1)
22.2 Basic probability exercises
392(1)
22.3 Branching processes
393(1)
22.4 The ballot problem and the hitting game
394(2)
22.5 Pascal's game
396(1)
22.6 Local Lemma
397(8)
III Solutions and hints to exercises
23 Solutions: Foundations
405(8)
23.1 Solutions: Properties of natural numbers
405(4)
23.2 Solutions: Well-ordered sets
409(1)
23.3 Solutions: Fermat's method of infinite descent
410(3)
24 Solutions: Inductive techniques applied to the infinite
413(2)
24.1 Solutions: More on well-ordered sets
413(1)
24.2 Solutions: Axiom of choice and equivalent forms
413(2)
25 Solutions: Paradoxes and sophisms
415(4)
25.1 Solutions: Trouble with the language?
415(1)
25.2 Solutions: Missed a case?
416(1)
25.3 Solutions: More deceit?
416(3)
26 Solutions: Empirical induction
419(6)
26.1 Solutions: Introduction
419(1)
26.2 Solutions: A Sequence of integers?
419(3)
26.3 Solutions: Sequences with only primes?
422(1)
26.4 Solutions: Divisibility
423(1)
26.5 Solutions: Never a Square?
423(1)
26.6 Solutions: Goldbach's conjecture
423(1)
26.7 Solutions: Cutting the Cake
424(1)
26.8 Solutions: Sums of hex numbers
424(1)
27 Solutions: Identities
425(90)
27.1 Solutions: Arithmetic progressions
425(35)
27.2 Solutions: Sums with binomial coefficients
460(22)
27.3 Solutions: Trigonometry
482(23)
27.4 Solutions: Miscellaneous identities
505(10)
28 Solutions: Inequalities
515(38)
29 Solutions: Number theory
553(54)
29.1 Solutions: Primes
553(9)
29.2 Solutions: Congruences
562(8)
29.3 Solutions: Divisibility
570(32)
29.4 Solutions: Expressible as sums
602(1)
29.5 Solutions: Egyptian fractions
603(1)
29.6 Solutions: Farey fractions
603(1)
29.7 Solutions: Continued fractions
603(4)
30 Solutions: Sequences
607(44)
30.1 Solutions: Difference sequences
607(2)
30.2 Solutions: Fibonacci numbers
609(26)
30.3 Solutions: Lucas numbers
635(3)
30.4 Solutions: Harmonic numbers
638(6)
30.5 Solutions: Catalan numbers
644(1)
30.6 Solutions: Eulerian numbers
645(1)
30.7 Solutions: Euler numbers
646(1)
30.8 Solutions: Stirling numbers
647(4)
31 Solutions: Sets
651(18)
31.1 Solutions: Properties of sets
651(9)
31.2 Solutions: Posets and lattices
660(2)
31.3 Solutions: Countable Zorn's lemma for measurable sets
662(1)
31.4 Solutions: Topology
663(5)
31.5 Solutions: Ultrafilters
668(1)
32 Solutions: Logic and language
669(4)
32.1 Solutions: Sentential logic
669(2)
32.2 Solutions: Well-formed formulae
671(2)
33 Solutions: Graphs
673(28)
33.1 Solutions: Graph theory basics
673(5)
33.2 Solutions: Trees and forests
678(4)
33.3 Solutions: Connectivity, walks
682(2)
33.4 Solutions: Matchings
684(1)
33.5 Solutions: Stable matchings
685(1)
33.6 Solutions: Graph coloring
686(3)
33.7 Solutions: Planar graphs
689(2)
33.8 Solutions: Extremal graph theory
691(6)
33.9 Solutions: Digraphs and tournaments
697(3)
33.10 Solutions: Geometric graphs
700(1)
34 Solutions: Recursion and algorithms
701(16)
34.1 Solutions: Recursively defined sets
701(1)
34.2 Solutions: Recursively defined sequences
702(13)
34.2.1 Solutions: Linear homogeneous recurrences of order 2
703(1)
34.2.2 Solutions: Applying the method of characteristic roots
703(3)
34.2.3 Solutions: Linear homogeneous recurrences of higher order
706(1)
34.2.4 Solutions: Non-homogeneous recurrences
707(2)
34.2.5 Solutions: Non-linear recurrences
709(5)
34.2.6 Solutions: Towers of Hanoi
714(1)
34.3 Solutions: Data structures
715(1)
34.4 Solutions: Complexity
716(1)
35 Solutions: Games and recreation
717(14)
35.1 Solutions: Tree games
717(5)
35.1.1 Solutions: The game of NIM
717(3)
35.1.2 Solutions: Chess
720(2)
35.2 Solutions: Dominoes and trominoes
722(3)
35.2.1 Solutions: Muddy children
724(1)
35.2.2 Solutions: Colored hats
725(1)
35.3 Solutions: Detecting counterfeit coin
725(6)
36 Solutions: Relations and functions
731(14)
36.1 Solutions: Binary relations
731(1)
36.2 Solutions: Functions
731(14)
37 Solutions: Linear and abstract algebra
745(34)
37.1 Solutions: Linear algebra
745(23)
37.2 Solutions: Groups and permutations
768(2)
37.3 Solutions: Rings
770(1)
37.4 Solutions: Fields
771(2)
37.5 Solutions: Vector spaces
773(6)
38 Solutions: Geometry
779(16)
38.1 Solutions: Convexity
781(2)
38.2 Solutions: Polygons
783(5)
38.3 Solutions: Lines, planes, regions, and polyhedra
788(5)
38.4 Solutions: Finite geometries
793(2)
39 Solutions: Ramsey theory
795(8)
40 Solutions: Probability and statistics
803(12)
IV Appendices
Appendix A ZFC axiom system
815(2)
Appendix B Inducing you to laugh?
817(4)
Appendix C The Greek alphabet
821(2)
References 823(42)
Name index 865(12)
Subject index 877
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.
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