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Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This textbook 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 familiarise 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 'scratchwork' sections to expose the machinery of proofs about the natural numbers, relations, functions and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. 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.

Tabla de contenidos provista por Syndetics

• 1 Sentential logic
• 2 Quantificational logic
• 3 Proofs
• 4 Relations
• 5 Functions
• 6 Mathematical induction
• 7 Infinite sets

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CHOICE Review

How To Prove It is a treatment of the fundamental ideas of logic, functions, and relations as understood by mathematicians for the purpose of writing correct proofs. This is material that every mathematics major needs to know. Often students first seriously confront the problem of constructing proofs in courses in discrete mathematics. Here, proof techniques and strategies are discussed at greater length and with more detailed examples than in discrete mathematics texts. Under the rubric of "structured proofs," a term coined by analogy with structured programming, Velleman thoroughly describes the process of assembling a proof piecemeal by formulating hypotheses, setting subgoals, and setting up logical alternatives. How To Prove It will be a useful supplement in discrete mathematics and other courses in which the art of proof is taught. Another book with the same purpose but different organization is Daniel Solow's How To Read and Do Proofs (1982; 2nd ed., 1990), which is arranged by proof technique, while Velleman's is arranged by mathematical topic. Both deserve space on library shelves. Undergraduate. M. Henle; Oberlin College