Handbook of mathematical induction: theory and applications
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton, FL [u.a.]
CRC Press
2011
|
| Schriftenreihe: | Discrete mathematics and its applications
58 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 823-863 |
| Beschreibung: | XXV, 893 S. graph. Darst. |
| ISBN: | 9781420093643 1420093649 |
Internformat
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| 245 | 1 | 0 | |a Handbook of mathematical induction |b theory and applications |c David S. Gunderson |
| 264 | 1 | |a Boca Raton, FL [u.a.] |b CRC Press |c 2011 | |
| 300 | |a XXV, 893 S. |b graph. Darst. | ||
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| 500 | |a Literaturverz. S. 823-863 | ||
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Datensatz im Suchindex
| _version_ | 1804140760070619136 |
|---|---|
| adam_text | Titel: Handbook of mathematical induction
Autor: Gunderson, David S
Jahr: 2011
Contents
Foreword xvii
Preface xix
About the author xxv
I Theory
1 What is mathematical induction? 1
1.1 Introduction............................... 1
1.2 An informal introduction to mathematical induction........ 2
1.3 Ingrcdients of a proof by mathematical induction.......... 3
1.4 Two other ways to think of mathematical induction......... 4
1.5 A simple example: Dice........................ 5
1.6 Gauss and sums............................. 6
1.7 A variety of applications........................ 9
1.8 History of mathematical induction.................. 11
1.9 Mathematical induction in modern literature ............ 15
2 Foundations 19
2.1 Notation................................. 19
2.2 Axioms ................................. 20
2.3 Peano s axioms............................. 22
2.4 Principle of mathematical induction ................. 23
2.5 Properties of natural numbers..................... 24
2.6 Well-ordered sets............................ 30
2.7 Well-founded sets............................ 33
3 Variants of finite mathematical induction 35
3.1 The first principle............................ 35
3.2 Strong mathematical induction.................... 36
3.3 Downward induction.......................... 38
vn
viii Contents
3.4 Alternative forms of mathematical induction ............. 42
3.5 Double induction............................ 43
3.6 Fermat s method of infinite descent................. . 46
3.7 Structural induction.......................... 48
4 Inductive techniques applied to the infinite 51
4.1 More on well-ordered sets....................... 51
4.2 Transfinite induction.......................... 53
4.3 Cardinais................................ 54
4.4 Ordinals................................. 55
4.5 Axiom of clioice and its equivalent forms............... 57
5 Paradoxes and sophisms from induction 69
5.1 Trouble with the language? ...................... 70
5.1.1 RicharcFs paradox........................ 70
5.1.2 Paradox of the unexpected exam................ 70
5.2 Fuzzy deflnitions............................ 71
5.2.1 No crowds allowed........................ 71
5.2.2 Nobody is rieh.......................... 71
5.2.3 Everyone is bald......................... 71
5.3 Missed a case? ............................. 71
5.3.1 All is for naught......................... 72
5.3.2 All horses are the same color.................. 72
5.3.3 Non-parallel lines go through one point ............ 72
5.4 More deeeit? .............................. 73
5.4.1 A new formula for triangulär numbers............. 73
5.4.2 All positive integers are equal.................. 74
5.4.3 Four weighings suffice...................... 74
6 Empirical induction 77
6.1 Introduction............................... 77
6.2 Guess the pattern?........................... 80
6.3 A pattern in primes?.......................... 80
6.4 A sequence of integers?......................... 80
6.5 Sequences with only primes?...................... 81
6.6 Divisibility................................ 82
6.7 Never a square?............................. 83
6.8 Goldbach s conjeeture ......................... 83
6.9 Cutting the cake............................ 83
6.10 Sums of hex numbers.......................... 84
6.11 Factoring xn ? 1 ............................ 85
6.12 Goodstein sequences.......................... 86
Contents ix
7 How to prove by induction 89
7.1 Tips on proving by induction..................... 89
7.2 Proving more can be easier ...................... 94
7.3 Proving limits by induction...................... 97
7.4 Whieh kind of induction is preferable?................ 103
7.4.1 When is induction needed?................... 103
7.4.2 Whieh kind of induction to use?................. 106
8 The written MI proof 109
8.1 A template ............................... 110
8.2 Improving the flow........................... 113
8.2.1 Using other results in a proof.................. 113
8.2.2 Clearly. it s trivial!........................ 114
8.2.3 Pronouns............................. 115
8.2.4 Footnotes............................. 115
8.2.5 We. let s. our. will. now. must.................. 115
8.3 Using notation and abbreviations................... 116
II Applications and exercises
9 Identities 125
9.1 Arithmetic progressions ........................ 125
9.2 Sums of finite geometric series and related series .......... 128
9.3 Power sums, sums of a Single power.................. 129
9.4 Products and sums of products.................... 131
9.5 Sums or products of fractions..................... 132
9.6 Identities with binomial coefficients.................. 134
9.6.1 Abel identities.......................... 141
9.6.2 Bernoulli numbers........................ 141
9.6.3 Faulhabers formula for power sums.............. 142
9.7 Gaussian coefficients.......................... 144
9.8 Trigonometry identities......................... 145
9.9 Miscellaneous identities ........................ 149
10 Inequalities 153
11 Number theory 161
11.1 Primes.................................. 161
11.2 Congruences............................... 167
11.3 Divisibility................................ 170
11.4 Numbers expressible as sums ..................... 176
11.5 Egyptian fractions........................... 176
x Contents
11.6 Farey fractions . ............................ 178
11.7 Continued fractions........................... 179
11.7.1 Finite continued fractions.................... 181
11.7.2 Infinite continued fractions................... 184
12 Sequences 187
12.1 Difference sequences.......................... 188
12.2 Fibonacci numbers........................... 190
12.3 Lucas numbers............................. 200
12.4 Harmonie munbers........................... 201
12.5 Catalan numbers............................ 203
12.5.1 Introduction ........................... 203
12.5.2 Catalan numbers defined by a formula............. 203
12.5.3 Cn as a number of ways to compute a produet........ 204
12.5.4 The definitions are equivalent.................. 205
12.5.5 Some oecurrenees of Catalan numbers............. 207
12.6 Schröder numbers............................ 208
12.7 Eulerian numbers............................ 209
12.7.1 Ascents. descents. rises. falls................... 209
12.7.2 Definitions for Eulerian numbers................ 210
12.7.3 Eulerian number exercises.................... 212
12.8 Euler numbers.............................. 213
12.9 Stirling numbers of the second kind.................. 214
13 Sets 217
13.1 Properties of sets............................ 217
13.2 Posets and lattices........................... 223
13.3 Topology................................. 226
13.4 Ultrafilters................................ 229
14 Logic and language 233
14.1 Sentential logic............................. 233
14.2 Equational logic............................. 235
14.3 Well-formed formulae.......................... 235
14.4 Language................................ 236
15 Graphs 239
15.1 Graph theory basics .......................... 239
15.2 Trees and forests............................ 242
15.3 Minimum spanning trees........................ 246
15.4 Connectivity, walks........................... 247
15.5 Matchings................................ 249
15.6 Stahle marriages ............................ 250
Contents xi
15.7 Graph coloring............................. 252
15.8 Planar gra])hs .............................. 253
15.9 Extremal graph tlieory......................... 255
15.10 Digraplis and tournanients....................... 257
15.11 Geometrie graphs............................ 258
16 Recursion and algorithms 261
10.1 Recursivoly dehned oporat iotis..................... 262
10.2 Recursivoly defined sets ........................ 2(i2
16.3 Recursively definod sequences..................... 263
16.3.1 Linear homogeneous recurrences of order 2 .......... 265
16.3.2 Method of eharacleristic roots.................. 200
16.3.3 Applying the method of eharacleristic roots.......... 209
16.3.4 Linear homogeneous recurrences of higher order........ 270
16.3.5 Non-homogeneous recurrences.................. 271
16.3.6 Finding lvcum iKrs ....................... 272
16.3.7 Non-linear recurrence ...................... 273
16.3.8 Towers of Hanoi ......................... 276
16.4 Loop invariants and algorithms.................... 277
16.5 Data struetures............................. 280
16.5.1 Gray codes............................ 281
16.5.2 The hypereube.......................... 281
16.5.3 Red-black trees.......................... 282
16.6 Complexity............................... 284
16.6.1 Landau notation......................... 28-1
16.6.2 The master theorem....................... 285
16.6.3 Closest pair of points....................... 286
17 Games and recreations 289
17.1 Introduction to game theory...................... 289
17.2 Tree games ............................... 290
17.2.1 Definitions and terminologv................... 290
17.2.2 The game of NIM ........................ 292
17.2.3 Chess............................... 293
17.3 Tiling with dominoes and trominoes................. 291
17.4 Dirty faces. cheating wives. muddy children. and colored hats . . . 295
17.4.1 A parlor game with sooty fingers................ 295
17.4.2 Unfaithful wives ......................... 296
17.4.3 The muddy children puzzle................... 298
17.4.4 Colored hats ........................... 299
17.4.5 More related puzzles and references .............. 300
17.5 Deteeting a counterfeit coin...................... 300
17.6 More recreations............................ 304
xü Contents
17.6.1 Pennies in boxes......................... 304
17.6.2 Josephus problem........................ 304
17.6.3 The gossip problem ....................... 307
17.6.4 Cars on a circular track..................... 307
18 Relations and functions 309
18.1 Binary relations............................. 309
18.2 Functions................................ 310
18.3 Calculus................................. 314
18.3.1 Derivatives............................ 314
18.3.2 Differential equations ...................... 315
18.3.3 Integration............................ 316
18.4 Polynomials............................... 320
18.5 Primitive recursive functions...................... 322
18.6 Ackermann s function ......................... 322
19 Linear and abstract algebra 325
19.1 Matrices and linear equations..................... 325
19.2 Groups and permutations....................... 334
19.2.1 Semigroups and groups ..................... 334
19.2.2 Permutations........................... 335
19.3 Rings................................... 338
19.4 Fields .................................. 338
19.5 Vector spaces.............................. 341
20 Geometry 349
20.1 Convexity................................ 350
20.2 Polygons................................. 354
20.3 Lines, planes, regions, and polyhedra................. 358
20.4 Finite geometries............................ 363
21 Ramsey theory 365
21.1 The Ramsey arrow........................... 366
21.2 Basic Ramsey theorems ........................ 367
21.3 Parameter words and combinatorial spaces.............. 373
21.4 Shelah bound.............................. 378
21.5 High chromatic number and large girth................ 383
22 Probability and statistics 387
22.1 Probability basics............................ 387
22.1.1 Probability spaces........................ 388
22.1.2 Independence and random variables.............. 389
22.1.3 Expected value and conditional probability.......... 390
Contents xiii
22.1.1 Conditional expectation..................... 391
22.2 Basic probability exercises....................... 392
22.3 Brandung processes .......................... 393
22.4 The ballot problem and the hitting game............... 391
22.5 Pascal s game.............................. 390
22.6 Local Lemma.............................. 397
III Solutions and hints to exercises
23 Solutions: Foundations 405
23.1 Solutions: Properties of natural numbers............... 105
23.2 Solutions: Well-ordered sets...................... 409
23.3 Solutions: Format s method of infinite deseent............ 410
24 Solutions: Inductive techniques applied to the infinite 413
24.1 Solutions: More on well-ordered sets................. 413
24.2 Solutions: Axiom of choice and equivalent forms .......... 413
25 Solutions: Paradoxes and sophisms 415
25.1 Solutions: Trouble with the language?................ 415
25.2 Solutions: Missed a case? ....................... 416
25.3 Solutions: More deceit? ........................ 416
26 Solutions: Empirical induction 419
26.1 Solutions: Introduction......................... 419
26.2 Solutions: A sequence of integers?................... 419
26.3 Solutions: Sequences with only primes?................ 422
26.4 Solutions: Divisibility ......................... 423
26.5 Solutions: Never a Square?....................... 423
26.6 Solutions: Goldbach s conjecture................... 423
26.7 Solutions: Cutting the cake...................... 424
26.8 Solutions: Sums of hex numbers.................... 424
27 Solutions: Identities 425
27.1 Solutions: Arithmetic progressions.................. 425
27.2 Solutions: Sums with binomial coefficients.............. 460
27.3 Solutions: Trigonometry........................ 482
27.4 Solutions: Miscellaneous identities .................. 505
28 Solutions: Inequalities 515
XIV
Contents
29 Solutions: Number theory 553
29.1 Solutions: Primes............................ 553
29.2 Solutions: Congruences.......................... 562
29.3 Solutions: Divisibility........................... 570
29.4 Solutions: Expressible as sums.................... 602
29.5 Solutions: Egyptian fractions..................... 603
29.6 Solutions: Farey fractions ....................... 603
29.7 Solutions: Continued fractions..................... 603
30 Solutions: Sequences 607
30.1 Solutions: Difference sequences.................... 607
30.2 Solutions: Fibonacci numbers..................... 609
30.3 Solutions: Lucas numbers....................... 635
30.4 Solutions: Harmonie numbers..................... 638
30.5 Solutions: Catalan numbers...................... 644
30.6 Solutions: Eulerian numbers...................... 645
30.7 Solutions: Euler numbers ....................... 646
30.8 Solutions: Stirling numbers...................... 647
31 Solutions: Sets 651
31.1 Solutions: Properties of sets...................... 651
31.2 Solutions: Posets and lattices..................... 660
31.3 Solutions: Countabie Zorn s lemma for measurable sets....... 662
31.4 Solutions: Topology .......................... 663
31.5 Solutions: Ultrafilters.......................... 668
32 Solutions: Logic and language 669
32.1 Solutions: Sentential logic....................... 669
32.2 Solutions: Well-formed formulae.................... 671
33 Solutions: Graphs 673
Graph theory basics.................... 673
Trees and forests...................... 678
Connectivity, walks..................... 682
Matchings.......................... 684
Stable matchings...................... 685
Graph coloring....................... 686
Planar graphs........................ 689
Extremal graph theory................... 691
Digraphs and tournaments................. 697
33.10 Solutions: Geometrie graphs...................... 700
33.1 Solutions
33.2 Solutions
33.3 Solutions
33.4 Solutions
33.5 Solutions
33.6 Solutions
33.7 Solutions
33.8 Solutions
33.9 Solutions
Contents xv
34 Solutions: Recursion and algorithms 701
34.1 Solutions: Reeursively defined sets.................. 701
34.2 Solutions: Recursively defined sequences............... 702
34.2.1 Solutions: Linear homogeneous recurrences of order 2 .... 703
34.2.2 Solutions: Applying the method of eharaeteristic roots .... 703
34.2.3 Solutions: Linear homogeneous recurrences of higher order . 706
34.2.4 Solutions: Non-homogeneous recurrences............ 707
34.2.5 Solutions: Non-linear recurrences................ 709
34.2.0 Solutions: Towers of Hanoi ................... 714
34.3 Solutions: Data structures....................... 715
34.4 Solutions: Complexity......................... 716
35 Solutions: Games and recreation 717
35.1 Solutions: Tree games......................... 717
35.1.1 Solutions: The game of NIM .................. 717
35.1.2 Solutions: Chess......................... 720
35.2 Solutions: Dominoes and trominoes.................. 722
35.2.1 Solutions: Muddy ehildren.................... 724
35.2.2 Solutions: Colored hats..................... 725
35.3 Solutions: Detecting counterfeit coin................. 725
36 Solutions: Relations and functions 731
36.1 Solutions: Binary relations....................... 731
36.2 Solutions: Functions.......................... 731
37 Solutions: Linear and abstract algebra 745
37.1 Solutions: Linear algebra........................ 745
37.2 Solutions: Groups and permutations................. 768
37.3 Solutions: Rings ............................ 770
37.4 Solutions: Fields............................ 771
37.5 Solutions: Vector spaces........................ 773
38 Solutions: Geometry 779
38.1 Solutions: Convexity.......................... 781
38.2 Solutions: Polygons........................... 783
38.3 Solutions: Lines, planes, regions. and polvhedra........... 788
38.4 Solutions: Finite geometries...................... 793
39 Solutions: Ramsey theory 795
40 Solutions: Probability and statistics 803
xvi Contents
IV Appendices
Appendix A: ZFC axiom System 815
Appendix B: Inducing you to laugh? 817
Appendix C: The Greek aiphabet 821
References 823
Name index 865
Subject index 877
|
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| spelling | Gunderson, David S. Verfasser (DE-588)114369666 aut Handbook of mathematical induction theory and applications David S. Gunderson Boca Raton, FL [u.a.] CRC Press 2011 XXV, 893 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications 58 Literaturverz. S. 823-863 Induction (Mathematics) Induktion (DE-588)4026765-9 gnd rswk-swf Induktion (DE-588)4026765-9 s DE-604 Discrete mathematics and its applications 58 (DE-604)BV023551867 58 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018668431&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gunderson, David S. Handbook of mathematical induction theory and applications Discrete mathematics and its applications Induction (Mathematics) Induktion (DE-588)4026765-9 gnd |
| subject_GND | (DE-588)4026765-9 |
| title | Handbook of mathematical induction theory and applications |
| title_auth | Handbook of mathematical induction theory and applications |
| title_exact_search | Handbook of mathematical induction theory and applications |
| title_full | Handbook of mathematical induction theory and applications David S. Gunderson |
| title_fullStr | Handbook of mathematical induction theory and applications David S. Gunderson |
| title_full_unstemmed | Handbook of mathematical induction theory and applications David S. Gunderson |
| title_short | Handbook of mathematical induction |
| title_sort | handbook of mathematical induction theory and applications |
| title_sub | theory and applications |
| topic | Induction (Mathematics) Induktion (DE-588)4026765-9 gnd |
| topic_facet | Induction (Mathematics) Induktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018668431&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023551867 |
| work_keys_str_mv | AT gundersondavids handbookofmathematicalinductiontheoryandapplications |