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| Preface | |
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| Part I | |
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| Language, Logic, and Sets | |
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| Logic and Language | |
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| Implication | |
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| Quantifiers and Definitions | |
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| Introduction to Sets | |
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| Introduction to Number Theory | |
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| Additional Set Theory | |
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| Definitions from Chapter 1 | |
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| Algebraic and Order Properties of Number Systems | |
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| Proofs | |
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| Proof Format I: Direct Proofs | |
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| Proof Format II: Contrapositive and Contradiction | |
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| Proof Format III: Existence, Uniqueness, Or | |
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| Proof Format IV: Mathematical Induction | |
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| The Fundamental Theorem of Arithmetic | |
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| Further Advice and Practice in Proving | |
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| Proof Formats | |
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| Functions | |
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| Definitions, Notation, and Examples | |
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| Composition, One-to-One, Onto, and Inverses | |
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| Images and Pre-Images of Sets | |
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| Definitions from Chapter 3 | |
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| Relations | |
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| Relations | |
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| Equivalence Relations | |
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| Partitions and Equivalence Relations | |
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| Partial Orders | |
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| Definitions from Chapter 4 | |
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| Part II | |
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| Infinite Sets | |
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| The Sizes of Sets | |
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| Countable Sets | |
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| Uncountable Sets | |
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| The Axiom of Choice and Its Equivalents | |
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| Definitions from Chapter 5 | |
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| Introduction to Discrete Mathematics | |
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| Graph Theory | |
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| Trees and Algorithms | |
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| Counting Principles I | |
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| Counting Principles II | |
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| Definitions from Chapter 6 | |
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| Introduction to Abstract Algebra | |
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| Operations and Properties | |
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| Groups | |
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| Groups in Geometry | |
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| Rings and Fields | |
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| Lattices | |
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| Homomorphisms | |
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| Definitions from Chapter 7 | |
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| Introduction to Analysis | |
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| Real Numbers, Approximations, and Exact Values | |
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| Zeno's Paradoxes | |
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| Limits of Functions | |
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| Continuous Functions and Counterexamples | |
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| Counterexamples in Rational Analysis | |
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| Sequences and Series | |
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| Discrete Dynamical Systems | |
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| The Intermediate Value Theorem | |
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| Definitions for Chapter 8 | |
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| Metamathematics and the Philosophy of Mathematics | |
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| Metamathematics | |
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| The Philosophy of Mathematics | |
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| Definitions for Chapter 9 | |
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| The Greek Alphabet | |
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| Answers: Selected Answers | |
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| Index | |
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| List of Symbols | |