Basic discrete mathematics: logic, set theory, & probability
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey
World Scientific
[2016]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xxvi, 706 Seiten Illustrationen, Diagramme |
| ISBN: | 9789814730396 9789813147546 |
Internformat
MARC
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| 245 | 1 | 0 | |a Basic discrete mathematics |b logic, set theory, & probability |c Richard Kohar (Royal Military College of Canada) |
| 264 | 1 | |a New Jersey |b World Scientific |c [2016] | |
| 300 | |a xxvi, 706 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Logic, Symbolic and mathematical | |
| 650 | 4 | |a Proof theory | |
| 650 | 4 | |a Induction (Mathematics) | |
| 650 | 4 | |a Set theory | |
| 650 | 4 | |a Probabilities | |
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Datensatz im Suchindex
| _version_ | 1804176733248684032 |
|---|---|
| adam_text | Contents
Preface xiü
Acknowledgments x*v
L Introduction to Logic i
1.1 Historical Development of Logic............................... 2
1.2 Propositions and Paradoxes.................................... 4
1.3 Connectives................................................. 8
1.3.1 Negation............................................ 9
1.3.2 Conjunction........................................... 9
1.3.3 Disjunction.......................................... 10
1.3.4 Exclusive Disjunction............................... 11
1.3.5 Implication.......................................... 11
1.3.6 Order of Precedence.................................. 13
1.4 Constructing Truth Tables for a Compound Proposition......... 15
1.5 Switching Circuits........................................... 17
1.6 Tautologies and Contradictions............................... 22
1.7 Logical Equivalence and Derived Logical Implications......... 23
1.7.1 Biconditional........................................ 23
1.7.2 Derived Logical Implications......................... 25
1.8 Review ...................................................... 28
1.9 Bibliographic Remarks........................................ 30
2. Proofs and Arguments 31
2.1 What is a Proof?............................................. 31
2.2 Rules of Inference........................................... 32
2.2.1 Fundamental Logical Equivalences..................... 32
2.2.2 Logical Equivalences for Implication and Biconditional 34
2.3 What is an Argument? ....................................... 39
vii
Basic Discrete Mathematics
viii
2.4 Modus Ponens.................................................. 44
2.5 Modus Tollens................................................. 46
2.6 Syllogism..................................................... 49
2.6.1 Hypothetical Syllogism................................ 49
2.6.2 Disjunctive Syllogism................................. 51
2.7 Fallacies..................................................... 53
2.7.1 Affirming the Disjunct................................ 54
2.7.2 Affirming the Consequent.............................. 55
2.7.3 Denying the Antecedent................................ 57
2.8 Deductive Reasoning and Inductive Reasoning................... 59
2.9 More Complex Arguments........................................ 60
2.10 Truth and Validity.......................................... 66
2.11 Review ....................................................... 71
2.12 Bibliographic Remarks......................................... 73
3. Sets and Set Operations 75
3.1 Introduction to Sets.......................................... 75
3.2 Russell’s Paradox............................................. 80
3.3 Set Operations................................................ 82
3.4 Principle of Inclusion and Exclusion.......................... 99
3.5 Set Operations Revisited......................................107
3.6 Using Venn Diagrams to Check Categorical Syllogisms...........Ill
3.7 Review .......................................................115
3.8 Bibliographic Remarks.........................................118
4. Infinity 119
4.1 Introduction..................................................119
4.1.1 Euclid and Prime Numbers .............................120
4.2 Galileo and Cantor Counting Numbers Leads to Heresy (or The
Integers and the Rationals are Countable) ....................121
4.3 Hilbert’s Hotel...............................................128
4.4 The Real Numbers are Not Countable............................133
4.5 Review .......................................................134
4.6 Bibliographic Remarks.........................................135
5. Elements of Combinatorics 137
5.1 The Pigeonhole Principle......................................137
5.2 Fundamental Counting Principles...............................139
5.2.1 The Multiplication Principle..........................139
5.2.2 The Sum Principle.....................................143
5.3 Factorials and Permutations...................................147
Contents ix
5.4 Permutations without Repetition................................151
5.5 Circular Permutations..........................................156
5.6 Permutations with Repetition...................................158
5.7 Combinations...................................................160
5.8 Permutations with Repetition of Indistinguishable Objects ... 174
5.9 Combinations with Repetition of Indistinguishable Objects: An
Application to Occupancy Problems..............................183
5.10 A Problem Solving Strategy for Counting Problems...............187
5.11 Review ........................................................192
5.12 Bibliographic Remarks..........................................197
6. Sequences and Series 199
6.1 Sequences......................................................199
6.2 Method of Finite Differences...................................212
6.3 Series and Sigma Notation......................................216
6.4 Arithmetic Series..............................................220
6.5 Properties of Series...........................................228
6.6 Pi Notation ...................................................235
6.7 The Geometric Series...........................................236
6.8 Investment: Applications of the Geometric Series...............242
6.8.1 Compounded Interest................................243
6.8.2 Annuities..........................................245
6.8.3 Present Value......................................246
6.9 The Infinite Geometric Series (A Trojan and a Tortoise)........249
6.10 Proofs by Mathematical Induction...............................256
6.10.1 Inductive Reasoning versus Mathematical Induction . . 257
6.10.2 The Principle of Mathematical Induction............257
6.10.3 Proofs Using the Principle of Mathematical Induction . 258
6.10.4 Abusing Mathematical Induction: More Fallacies . . . . 262
6.10.5 Guessing the Formula...............................265
6.11 Review ........................................................272
6.12 Bibliographic Remarks..........................................275
7. The Binomial Theorem 277
7.1 Pascal’s Triangle .............................................277
7.2 The Binomial Theorem...........................................291
7.3 Review ....................................................... 302
7.4 Bibliographic Remarks........................................ 305
8. Introduction to Probability 307
8.1 Introduction...................................................307
X
Basic Discrete Mathematics
8.2 Experiments, Sample Spaces, Sample Points, and Events .... 307
8.3 Probability of an Event in a Uniform Sample Space..............315
8.4 Definition and Axioms of Probability...........................320
8.5 Simple Probability Rules.......................................323
8.6 Risk: Applications of Probability..............................328
8.7 Using Counting Techniques to Calculate Probability.............334
8.7.1 Permutations...........................................335
8.7.2 Combinations...........................................336
8.8 Conditional Probability........................................341
8.9 Independence...................................................354
8.9.1 Independence versus Mutually Exclusive Events .... 359
8.9.2 The Sally Clark Case ..................................360
8.10 The Origins of Probability: Gambling...........................367
8.11 Bayes’ Theorem.................................................371
8.12 Review ........................................................388
8.13 Bibliographic Remarks..........................................390
9. Random Variables 393
9.1 Introduction...................................................393
9.2 Random Variables Probability Distributions...................393
9.3 Expected Value.................................................401
9.4 Variance Standard Deviation..................................408
9.5 Review ........................................................414
10. Probability Distributions 417
10.1 Uniform Distribution...........................................417
10.2 Binomial Distribution..........................................421
10.3 The Poisson Distribution.......................................435
10.4 The Geometric Distribution.....................................449
10.5 The Hypergeometric Distribution................................455
10.6 The Normal (or Gaussian) Distribution: A Bridge from Discrete
to Continuous Mathematics .....................................465
10.7 Review ........................................................475
10.8 Bibliographic Remarks..........................................478
Appendix A Probability Distribution Tables 479
A. 1 Standard Normal Distribution Table.............................480
Appendix B Prerequisite Knowledge 483
B. l Number Systems.................................................483
B.2 Geometric Formulas.............................................483
Contents xi
B.3 Basic Properties of Real Numbers.................................485
B.4 Solving Quadratic Equations......................................489
B.5 Exponent Laws....................................................490
B.6 Logarithms ......................................................491
B.7 Scientific Notation ...........................................494
Appendix C Solutions to Exercises 497
Bibliography 689
Index 699
Basic Discrete Mathematics
Logic, SetTheory, Probability
This lively introductory text exposes the student in the humanities to the world of
discrete mathematics. A problem-solving based approach grounded in the ideas
of George Pólya are at the heart of this book. Students learn to handle and solve
new problems on their own. A straightforward, clear writing style, and well-crafted
examples with diagrams invite the students to develop into precise and critical
thinkers. Particular attention has been given to the material that some students find
challenging, such as proofs, This book illustrates how to spot invalid arguments, to
enumerate possibilities, and to construct probabilities. It also presents case studies
to students about the possible detrimental effects of ignoring these basic principles.
The book is invaluable for a discrete and finite mathematics course at the freshman
undergraduate level or for self-study since there are full solutions to the exercises
in an appendix.
|
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| indexdate | 2024-07-10T07:36:52Z |
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| isbn | 9789814730396 9789813147546 |
| language | English |
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| physical | xxvi, 706 Seiten Illustrationen, Diagramme |
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| spelling | Kohar, Richard 1988- Verfasser (DE-588)1119749298 aut Basic discrete mathematics logic, set theory, & probability Richard Kohar (Royal Military College of Canada) New Jersey World Scientific [2016] xxvi, 706 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Logic, Symbolic and mathematical Proof theory Induction (Mathematics) Set theory Probabilities Diskrete Mathematik (DE-588)4129143-8 gnd rswk-swf Diskrete Mathematik (DE-588)4129143-8 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029266540&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029266540&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Kohar, Richard 1988- Basic discrete mathematics logic, set theory, & probability Logic, Symbolic and mathematical Proof theory Induction (Mathematics) Set theory Probabilities Diskrete Mathematik (DE-588)4129143-8 gnd |
| subject_GND | (DE-588)4129143-8 |
| title | Basic discrete mathematics logic, set theory, & probability |
| title_auth | Basic discrete mathematics logic, set theory, & probability |
| title_exact_search | Basic discrete mathematics logic, set theory, & probability |
| title_full | Basic discrete mathematics logic, set theory, & probability Richard Kohar (Royal Military College of Canada) |
| title_fullStr | Basic discrete mathematics logic, set theory, & probability Richard Kohar (Royal Military College of Canada) |
| title_full_unstemmed | Basic discrete mathematics logic, set theory, & probability Richard Kohar (Royal Military College of Canada) |
| title_short | Basic discrete mathematics |
| title_sort | basic discrete mathematics logic set theory probability |
| title_sub | logic, set theory, & probability |
| topic | Logic, Symbolic and mathematical Proof theory Induction (Mathematics) Set theory Probabilities Diskrete Mathematik (DE-588)4129143-8 gnd |
| topic_facet | Logic, Symbolic and mathematical Proof theory Induction (Mathematics) Set theory Probabilities Diskrete Mathematik |
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