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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 | |