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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...
