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Preface | |
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Logic | |
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Statements, Negation, and Compound Statements | |
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Truth Tables and Logical Equivalences | |
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Conditional and Biconditional Statements | |
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Logical Arguments | |
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Open Statements and Quantifiers | |
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Chapter Review | |
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Deductive Mathematical Systems and Proofs | |
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Deductive Mathematical Systems | |
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Mathematical Proofs | |
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Techniques for Proving the Conditional Statement P → Q | |
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Additional Proof Techniques | |
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Conjectures, Proofs, and Disproofs | |
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The System of Rational Numbers and the System of Real Numbers | |
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Chapter Review | |
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Set Theory | |
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Sets and Subsets | |
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Set Operations | |
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Additional Set Operations | |
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Generalized Set Union and Intersection | |
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Chapter Review | |
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Relations | |
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Relations | |
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The Order Relations <, ≤, >, ≥ | |
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Reflexive, Symmetric, Transitive, and Equivalence Relations | |
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Equivalence Relations, Equivalence Classes, and Partitions | |
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Chapter Review | |
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Functions | |
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Functions | |
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Onto Functions, One-to-One Functions, and One-to-One Correspondences | |
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Inverse of a Function | |
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Images and Inverse Images of Sets | |
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Chapter Review | |
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Mathematical Induction | |
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Mathematical Induction | |
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The Well-Ordering Principle and the Fundamental Theorem of Arithmetic | |
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Cardinalities of Sets | |
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Finite Sets | |
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Denumerable and Countable Sets | |
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Uncountable Sets | |
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Proofs from Real Analysis | |
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Sequences | |
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Limit Theorems for Sequences | |
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Monotone Sequences and Subsequences | |
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Cauchy Sequences | |
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Proofs from Group Theory | |
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Binary Operations and Algebraic Structures | |
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Groups | |
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Subgroups and Cyclic Groups | |
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A Reading and Writing Mathematical Proofs | |
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Answers to Selected Exercises | |
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References | |
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Index | |