Mathematics: a discrete introduction
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Belmont, Calif.
Cengage Learning
2013
|
| Ausgabe: | 3. ed., international edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 470 S. |
| ISBN: | 9780840065285 |
Internformat
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| 245 | 1 | 0 | |a Mathematics |b a discrete introduction |c Edward R. Scheinerman |
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| 264 | 1 | |a Belmont, Calif. |b Cengage Learning |c 2013 | |
| 300 | |a XXI, 470 S. | ||
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Datensatz im Suchindex
| _version_ | 1804151348806025216 |
|---|---|
| adam_text | Contents
To the Student xvii
How to Read a Mathematics Book xviii
Exercises xviii
To the Instructor xix
Audience and Prerequisites xix
Topics Covered and Navigating the Sections xix
Sample Course Outlines xx
Special Features xx
What s New in this Third Edition xxiii
Acknowledgments xxv
This New Edition xxv
From the Second Edition xxv
From the First Edition xxv
1 Fundamentals 1
1 Joy 1
Why? 1
The Agony and the Ecstasy 1
Exercise 2
2 Speaking (and Writing) of Mathematics 2
Precisely! 2
A Bit of Help 2
Exercise 3
3 Definition 4
Recap 6
Exercises 6
4 Theorem 8
The Nature of Truth 8
If-Then 9
If and Only If 11
And, Or, and Not 12
What Theorems Are Called 12
Vacuous Truth 13
Recap 13
Exercises 13
5 Proof 15
A More Involved Proof 18
Proving If-and-Only-If Theorems 20
Proving Equations and Inequalities 22
Recap 22
Exercises 22
6 Counterexample 23
Recap 24
Exercises 24
VII
Contents
7 Boolean Algebra 25
More Operations 27
Recap 28
Exercises 28
Chapter 1 Self Test 30
2 Collections 33
8 Lists 33
Counting Two-Element Lists 33
Longer Lists 35
Recap 38
Exercises 38
9 Factorial 40
Much Ado About 0! 40
Product Notation 41
Recap 42
Exercises 42
10 Sets I: Introduction, Subsets 43
Equality of Sets 45
Subset 46
Counting Subsets 48
Power Set 49
Recap 49
Exercises 50
11 Quantifiers 51
There Is 51
For All 52
Negating Quantified Statements 53
Combining Quantifiers 53
Recap 54
Exercises 54
12 Sets II: Operations 56
Union and Intersection 56
The Size of a Union 58
Difference and Symmetric Difference 60
Cartesian Product 64
Recap 64
Exercises 64
13 Combinatorial Proof: Two Examples
Recap 69
Exercises 69
Chapter 2 Self Test 70
3 Counting and Relations 73
14 Relations 73
Properties of Relations 75
Recap 76
Exercises 76
Contents
IX
15 Equivalence Relations 78
Equivalence Classes 81
Recap 83
Exercises 83
16 Partitions 85
Counting Classes/Parts 86
Recap 88
Exercises 89
17 Binomial Coefficients 90
Calculating (nk) 93
Pascal’s Triangle 94
A Formula for ( ) 96
Counting Lattice Paths 97
Recap 98
Exercises 98
18 Counting Multisets 101
Multisets 101
Formulas for ((JJ)) 103
Extending the Binomial Theorem to Negative Powers 105
Recap 107
Exercises 108
19 Inclusion-Exclusion 109
How to Use Inclusion-Exclusion 112
Derangements 114
A Ghastly Formula 116
Recap 116
Exercises 116
Chapter 3 Self Test 117
4 More Proof 119
20 Contradiction 119
Proof by Contrapositive 119
Reductio Ad Absurdum 120
Proof by Contradiction and Sudoku 123
A Matter of Style 124
Recap 124
Exercises 124
21 Smallest Counterexample 125
Well-Ordering 129
Recap 134
Exercises 134
And Finally 134
22 Induction 135
The Induction Machine 135
Theoretical Underpinnings 136
Proof by Induction 137
Proving Equations and Inequalities 139
Other Examples 140
Strong Induction 141
A More Complicated Example 143
A Matter of Style 145
Recap 145
Exercises 145
23 Recurrence Relations 149
First-Order Recurrence Relations 150
Second-Order Recurrence Relations 153
The Case of the Repeated Root 155
Sequences Generated by Polynomials 157
Recap 163
Exercises 163
Chapter 4 Self Test 165
Functions 167
24 Functions 167
Domain and Image 168
Pictures of Functions 170
Counting Functions 170
Inverse Functions 171
Counting Functions, Again 174
Recap 175
Exercises 175
25 The Pigeonhole Principle 178
Cantor’s Theorem 180
Recap 181
Exercises 181
26 Composition 183
Identity Function 186
Recap 186
Exercises 186
27 Permutations 188
Cycle Notation 189
Calculations with Permutations 191
Transpositions 192
A Graphical Approach 196
Recap 198
Exercises 198
28 Symmetry 200
Symmetries of a Square 200
Symmetries as Permutations 201
Combining Symmetries 202
Formal Definition of Symmetry 203
Recap 204
Exercises 204
29 Assorted Notation 204
Big Oh 205
Sl and 0 207
Little Oh 207
Floor and Ceiling 208
/,/(*), and/(•) 208
Recap 209
Exercises 209
Chapter 5 Self Test
210
Contents
XI
6 Probability 213
30 Sample Space 213
Recap 215
Exercises 215
31 Events 217
Combining Events 218
The Birthday Problem 220
Recap 221
Exercises 221
32 Conditional Probability and Independence 223
Independence 224
Independent Repeated Trials 226
The Monty Hall Problem 227
Recap 227
Exercises 228
33 Random Variables 231
Random Variables as Events 232
Independent Random Variables 233
Recap 234
Exercises 234
34 Expectation 235
Linearity of Expectation 238
Product of Random Variables 241
Expected Value as a Measure of Centrality 243
Variance 244
Recap 247
Exercises 248
Chapter 6 Self Test 250
7 Number Theory 253
35 Dividing 253
Div and Mod 255
Recap 256
Exercises 256
36 Greatest Common Divisor 258
Calculating the gcd 258
Correctness 260
How Fast? 261
An Important Theorem 262
Recap 264
Exercises 264
37 Modular Arithmetic 266
A New Context for Basic Operations 266
Modular Addition and Multiplication 267
Modular Subtraction 268
Modular Division 269
A Note on Notation 273
Recap 273
Exercises 273
XII
Contents
38 The Chinese Remainder Theorem 275
Solving One Equation 275
Solving Two Equations 276
Recap 278
Exercises 278
39 Factoring 279
Infinitely Many Primes 280
A Formula for Greatest Common Divisor 281
Irrationality of fl 281
Just for Fun 283
Recap 283
Exercises 283
Chapter 7 Self Test 287
8 Algebra 289
40 Groups 289
Operations 289
Properties of Operations 290
Groups 291
Examples 293
Recap 295
Exercises 295
41 Group Isomorphism 297
The Same? 297
Cyclic Groups 299
Recap 301
Exercises 301
42 Subgroups 302
Lagrange’s Theorem 304
Recap 307
Exercises 307
43 Fermat s Little Theorem 309
First Proof 309
Second Proof 310
Third Proof 313
Euler’s Theorem 313
Primality Testing 314
Recap 315
Exercises 315
44 Public Key Cryptography I: Introduction 316
The Problem: Private Communication in Public 316
Factoring 316
Words to Numbers 317
Cryptography and the Law 318
Recap 318
Exercises 319
45 Public Key Cryptography II: Rabin s Method 319
Square Roots Modulo n 319
The Encryption and Decryption Procedures 323
Recap 323
Exercises 323
Contents
xiii
46 Public Key Cryptography III: RSA 325
The RSA Encryption and Decryption Functions 325
Security 327
Recap 328
Exercises 328
Chapter 8 Self Test 329
9 Graphs 331
47 Fundamentals of Graph Theory 331
Map Coloring 331
Three Utilities 332
Seven Bridges 333
What Is a Graph? 333
Adjacency 334
A Matter of Degree 335
Further Notation and Vocabulary 336
Recap 337
Exercises 337
48 Subgraphs 339
Induced and Spanning Subgraphs 339
Cliques and Independent Sets 341
Complements 342
Recap 343
Exercises 343
49 Connection 344
Walks 345
Paths 345
Disconnection 348
Recap 349
Exercises 349
50 Trees 350
Cycles 350
Forests and Trees 350
Properties of Trees 351
Leaves 352
Spanning Trees 354
Recap 354
Exercises 355
51 Eulerian Graphs 356
Necessary Conditions 357
Main Theorems 358
Unfinished Business 359
Recap 360
Exercises 360
52 Coloring 361
Core Concepts 361
Bipartite Graphs 363
The Ease of Two-Coloring and the Difficulty of Three-Coloring 365
Recap 366
Exercises 366
XIV
Contents
53 Planar Graphs 367
Dangerous Curves 367
Embedding 368
Euler’s Formula 368
Nonplanar Graphs 371
Coloring Planar Graphs 372
Recap 374
Exercises 374
Chapter 9 Self Test 375
10 Partially Ordered Sets 379
54 Fundamentals of Partially Ordered Sets 379
What is a Poset? 379
Notation and Language 381
Recap 382
Exercises 382
55 Max and Min 384
Recap 385
Exercises 385
56 Linear Orders 386
Recap 388
Exercises 388
57 Linear Extensions 389
Sorting 391
Linear Extensions of Infinite Posets 393
Recap 393
Exercises 394
58 Dimension 394
Realizers 394
Dimension 396
Embedding 398
Recap 400
Exercises 400
59 Lattices 400
Meet and Join 400
Lattices 402
Recap 404
Exercises 404
Chapter 10 Self Test 405
Appendices 409
A Lots of Hints and Comments; Some Answers 409
Contents
xv
B Solutions to Self
Chapter 1 435
Chapter 2 436
Chapter 3 437
Chapter 4 438
Chapter 5 441
Chapter 6 443
Chapter 7 446
Chapter 8 447
Chapter 9 450
Chapter 10 453
C Glossary 456
D Fundamentals
Numbers 462
Operations 462
Ordering 462
462
Complex Numbers
Substitution 463
463
435
Index 465
|
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| indexdate | 2024-07-10T00:53:24Z |
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| isbn | 9780840065285 |
| language | English |
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| spelling | Scheinerman, Edward Verfasser (DE-588)113762938X aut Mathematics a discrete introduction Edward R. Scheinerman 3. ed., international edition Belmont, Calif. Cengage Learning 2013 XXI, 470 S. txt rdacontent n rdamedia nc rdacarrier Mathematik (DE-588)4037944-9 gnd rswk-swf Informatik (DE-588)4026894-9 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Informatik (DE-588)4026894-9 s Mathematik (DE-588)4037944-9 s DE-604 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026735508&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Scheinerman, Edward Mathematics a discrete introduction Mathematik (DE-588)4037944-9 gnd Informatik (DE-588)4026894-9 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4026894-9 (DE-588)4151278-9 |
| title | Mathematics a discrete introduction |
| title_auth | Mathematics a discrete introduction |
| title_exact_search | Mathematics a discrete introduction |
| title_full | Mathematics a discrete introduction Edward R. Scheinerman |
| title_fullStr | Mathematics a discrete introduction Edward R. Scheinerman |
| title_full_unstemmed | Mathematics a discrete introduction Edward R. Scheinerman |
| title_short | Mathematics |
| title_sort | mathematics a discrete introduction |
| title_sub | a discrete introduction |
| topic | Mathematik (DE-588)4037944-9 gnd Informatik (DE-588)4026894-9 gnd |
| topic_facet | Mathematik Informatik Einführung |
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