First Course in Mathematical Logic and Set Theory

Informationen zum Autor Michael L. O'Leary, PhD, is Professor of Mathematics at the College of DuPage in Glen Ellyn, Illinois. He received his doctoral degree in mathematics from the University of California, Irvine in 1994 and is the author of Revolutions of Geometry, also published by Wiley. Klappentext A mathematical introduction to the theory and applications of logic and set theory with an emphasis on writing proofsHighlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, A First Course in Mathematical Logic and Set Theory introduces how logic is used to prepare and structure proofs and solve more complex problems.The book begins with propositional logic, including two-column proofs and truth table applications. This is followed by first-order logic, which provides the structure for writing mathematical proofs. Set theory is then introduced and serves as the basis for defining relations, functions, numbers, mathematical induction, ordinals, and cardinals. The book concludes with a primer on basic model theory with applications to abstract algebra. A First Course in Mathematical Logic and Set Theory also includes:* Section exercises designed to show the interactions between topics and reinforce the presented ideas and concepts* Numerous examples that illustrate theorems and employ basic concepts such as Euclid's Lemma, the Fibonacci sequence, and unique factorization* Coverage of important theorems including the well-ordering theorem, completeness theorem, compactness theorem, as well as the theorems of Löwenheim-Skolem, Burali and Forti, Hartog, Cantor-Schröeder-Bernstein, and KönigAn excellent textbook for students studying the foundations of mathematics and mathematical proofs, A First Course in Mathematical Logic and Set Theory is also appropriate for readers preparing for careers in mathematics education or computer science. In addition, the book is ideal for introductory courses on mathematical logic and/or set theory and appropriate for upper-undergraduate level transition courses with rigorous mathematical reasoning involving algebra, number theory, or analysis.Michael L. O'Leary, PhD, is Professor of Mathematics at the College of DuPage in Glen Ellyn, Illinois. He received his doctoral degree in mathematics from the University of California, Irvine in 1994 and is the author of Revolutions of Geometry, also published by Wiley. Zusammenfassung Rather than teach mathematics and the structure of proofssimultaneously, this book first introduces logic as the foundationof proofs and then demonstrates how logic applies to mathematicaltopics. This method ensures that readers gain a firmunderstanding of how logic interacts with mathematics and empowersthem to solve more complex problems. Inhaltsverzeichnis Preface xiiiAcknowledgments xvList of Symbols xvii1 Propositional Logic 11.1 Symbolic Logic 1Propositions 2Propositional Forms 6Interpreting Propositional Forms 8Valuations and Truth Tables 111.2 Inference 20Semantics 22Syntactics 241.3 Replacement 32Semantics 32Syntactics 351.4 Proof Methods 41Deduction Theorem 41Direct Proof 46Indirect Proof 481.5 The Three Properties 53Consistency 53Soundness 57Completeness 602 FirstOrderLogic 652.1 Languages 65Predicates 65Alphabets 69Terms 72Formulas 732.2 Substitution 77Terms 77Free Variables 79Formulas 802.3 Syntactics 87Quantifier Negation 87Proofs with Universal Formulas 89Proofs with Existential Formulas 932.4 Proof Methods 98Universal Proofs 100Existential Proofs 101Multiple Quantifiers 103Counterexamples 104Direct Proof 105Existence and Uniqueness 107Indirect Proof 108Biconditional Proof 110Proof of Disunctions 114Proof by Cases 1143 Set Theory 1193.1 Sets and Elements 119Rosters 120Famous Sets 121Abstraction 1233.2 Set Operations 128Union and Intersection...

List of contents

Preface xiii
 
Acknowledgments xv
 
List of Symbols xvii
 
1 Propositional Logic 1
 
1.1 Symbolic Logic 1
 
Propositions 2
 
Propositional Forms 6
 
Interpreting Propositional Forms 8
 
Valuations and Truth Tables 11
 
1.2 Inference 20
 
Semantics 22
 
Syntactics 24
 
1.3 Replacement 32
 
Semantics 32
 
Syntactics 35
 
1.4 Proof Methods 41
 
Deduction Theorem 41
 
Direct Proof 46
 
Indirect Proof 48
 
1.5 The Three Properties 53
 
Consistency 53
 
Soundness 57
 
Completeness 60
 
2 FirstOrder
 
Logic 65
 
2.1 Languages 65
 
Predicates 65
 
Alphabets 69
 
Terms 72
 
Formulas 73
 
2.2 Substitution 77
 
Terms 77
 
Free Variables 79
 
Formulas 80
 
2.3 Syntactics 87
 
Quantifier Negation 87
 
Proofs with Universal Formulas 89
 
Proofs with Existential Formulas 93
 
2.4 Proof Methods 98
 
Universal Proofs 100
 
Existential Proofs 101
 
Multiple Quantifiers 103
 
Counterexamples 104
 
Direct Proof 105
 
Existence and Uniqueness 107
 
Indirect Proof 108
 
Biconditional Proof 110
 
Proof of Disunctions 114
 
Proof by Cases 114
 
3 Set Theory 119
 
3.1 Sets and Elements 119
 
Rosters 120
 
Famous Sets 121
 
Abstraction 123
 
3.2 Set Operations 128
 
Union and Intersection 128
 
Set Difference 129
 
Cartesian Products 132
 
Order of Operations 134
 
3.3 Sets within Sets 137
 
Subsets 137
 
Equality 139
 
3.4 Families of Sets 150
 
Power Set 153
 
Union and Intersection 154
 
Disjoint and Pairwise Disjoint 157
 
4 Relations and Functions 163
 
4.1 Relations 163
 
Composition 165
 
Inverses 167
 
4.2 Equivalence Relations 170
 
Equivalence Classes 173
 
Partitions 175
 
4.3 Partial Orders 179
 
Bounds 183
 
Comparable and Compatible Elements 184
 
WellOrdered Sets 186
 
4.4 Functions 192
 
Equality 197
 
Composition 198
 
Restrictions and Extensions 200
 
Binary Operations 200
 
4.5 Injections and Surjections 207
 
Injections 208
 
Surjections 211
 
Bijections 214
 
Order Isomorphims 215
 
4.6 Images and Inverse Images 220
 
5 Axiomatic Set Theory 227
 
5.1 Axioms 227
 
Equality Axioms 228
 
Existence and Uniqueness Axioms 229
 
Construction Axioms 230
 
Replacement Axioms 231
 
Axiom of Choice 232
 
Axiom of Regularity 236
 
5.2 Natural Numbers 239
 
Order 241
 
Recursion 244
 
Arithmetic 245
 
5.3 Integers and Rational Numbers 251
 
Integers 252
 
Rational Numbers 255
 
Actual Numbers 258
 
5.4 Mathematical Induction 259
 
Combinatorics 263
 
Euclid?s Lemma 267
 
5.5 Strong Induction 270
 
Fibonacci Sequence 271
 
Unique Factorization 273
 
5.6 Real Numbers 277
 
Dedekind Cuts 278
 
Arithmetic 280
 
Complex Numbers 283
 
6 Ordinals and Cardinals 285
 
6.1 Ordinal Numbers 285
 
Ordinals 288
 
Classification 292
 
BuraliForti and Hartogs 294
 
Transfinite

Product details

• Authors: &apos, Michael L. leary, O&, O&apos, Michael Oleary, O'Leary, Michael O'Leary, Michael L O'Leary, Michael L. O'Leary, Michael L. (College of DuPage) O'Leary, Michael O''leary, Michael L. O''leary, Michael L. (College of Dupage) O''leary, ML O''leary, O'Leary Michael L.
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 16.10.2015
• EAN: 9780470905883
• ISBN: 978-0-470-90588-3
• No. of pages: 464
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Basic principles; Mathematik, Philosophie, Logik, Philosophy, Diskrete Mathematik, MATHEMATICS / Logic, Mathematics, Mathematische Logik, philosophische Logik, Discrete Mathematics, Logik u. Grundlagen der Mathematik, Logic & Foundations, Philosophical Logic