## A Logical Introduction to ProofThe book is intended for students who want to learn how to prove theorems and be better prepared for the rigors required in more advance mathematics. One of the key components in this textbook is the development of a methodology to lay bare the structure underpinning the construction of a proof, much as diagramming a sentence lays bare its grammatical structure. Diagramming a proof is a way of presenting the relationships between the various parts of a proof. A proof diagram provides a tool for showing students how to write correct mathematical proofs. |

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### Contents

Chapter 1 Propositional Logic
| 1 |

Chapter 2 Predicate Logic
| 29 |

Chapter 3 Proof Strategies and Diagrams
| 61 |

Chapter 4 Mathematical Induction
| 99 |

Chapter 5 Set Theory
| 143 |

Chapter 6 Functions
| 169 |

Chapter 7 Relations
| 209 |

Chapter 8 Core Concepts in Abstract Algebra
| 239 |

Chapter 9 Core Concepts in Real Analysis
| 293 |

Appendix Summary of Strategies
| 341 |

References | 347 |

349 | |

351 | |

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algebraic structure argument Assume Assumption Strategy axiom of choice Base step binary operation conclude congruence relation contradiction countable cycle D I R deﬁned Deﬁnition equation equivalence relation Example false family of sets ﬁnd ﬁrst formula function f given Hence identity element implies indexed family Inductive step integers inverse Lemma Let f Let G LetA lim sn logical form logically equivalent lower bound mathematical induction mathematical proofs means natural number negation laws nonzero normal subgroup obtain permutation positive real numbers prime number proof diagram proof of Theorem Proof Strategy ProofAnalysis propositional logic propositional sentence quantiﬁer quantifiers rational numbers real numbers ring satisfies sequence Solution statement subgroup of G subset Suppose Theorem true truth table truth values upper bound well-ordering principle write xP(x