## How to Prove It: A Structured ApproachMany students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians. |

### Contents

0521861241int_p1 | 1 |

0521861241c01_p854 | 8 |

0521861241c02_p5583 | 55 |

0521861241c03_p84162 | 84 |

0521861241c04_p163225 | 163 |

0521861241c05_p226259 | 226 |

0521861241c06_p260305 | 260 |

0521861241c07_p306328 | 306 |

0521861241apx1_p329372 | 329 |

0521861241apx2_p373374 | 373 |

375 | |

0521861241sum_p376 | 376 |

381 | |

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### Common terms and phrases

A X B Analyze the logical arbitrary element assume assumption Chapter choose conclude countable counterexample deﬁned deﬁnition disjoint element of F equation equivalence classes equivalence relation example exercise existential f is one-to-one fact false family of sets Figure ﬁnd ﬁnite ﬁrst following proof following statements form xP(x free variables function f Givens Goal Hint Induction step inductive hypothesis Let f logical forms mathematical induction mathematicians means minimal element modus ponens natural number negation law notation ordered pairs partial order plug positive integer prime numbers proof by contradiction Proof Designer proof strategies prove a goal quantiﬁer quantifiers real number recursive reexpress reﬂexive Scratch Similarly smallest element Solution Theorem stand statement P(x strong induction subset Suppose f symbols symmetric closure Theorem total order transitive closure truth set truth table universe of discourse Venn diagrams Vx G words write