Proof, Logic, and Conjecture: The Mathematician’s Toolbox by Robert S. Wolf (PDF)

Description

Starting with an explanation of what ‘proof’ means to a mathematician, this student-friendly introductory text is aimed at undergraduates in mathematics and related disciplines. It shows students how to read and write mathematical proofs and describes how mathematicians investigate problems and formulate conjecture. Students develop their skills in logic by following precise rules, and examples and exercises relating to discovery and conjecture appear throughout. The author also covers mathematical concepts such as real and complex numbers, relations and functions and set theory.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I cannot BELIEVE this book is not 5 Stars. This is the most complete book of its kind. I used this book for my “introduction to proofs” class when I was an undergrad and I can tell you this book has played a large part in my mathematical life. Before you can learn mathematics, you really need to know how to prove things at a fundamental level. This book will give your students the proper tools to enter upper division mathematics. I must say that when i took this class with the author it was, and is still, one of the hardest classes I have every had, however, I would not have made it to graduate school otherwise. The only books I have brought to grad school have been Analysis by W. Rudin and This book. If you think you want to be a Math Major, do yourself a favor and buy this book……

⭐Mathematics consists of calculations and proofs. Elementary mathematics consists mainly of calculations, and students often have difficulty when they advance to the point where proofs become important. This book is intended to help students develop the “mathematical sophistication” they will need in advanced courses. That sophistication involves concepts and skills from three areas: logic and proofs, sets and functions, and the basic number systems used in mathematics.The first four chapters of the book are devoted to logic and proofs; the next three to sets and functions; and the last three to number systems. Everything that should be in such a book is included: propositional and predicate logic, proof by induction, Russell’s paradox, functions as sets of ordered pairs, the concept of cardinality; examples of rings and fields, the completeness axiom for the real numbers; complex numbers as pairs of real numbers.Dr. Wolf has brought to this book a lively wit, twenty years of teaching experience with the target audience, and the acumen and scholarship of a highly-trained mathematician and logician. The book thus entertains and educates, without sacrificing accuracy or precision. The twenty years of experience, for example, is highly visible in the section on “Hints for Finding Proofs”. The scholarship is visible in the “Suggestions for Further Reading” at the end of each chapter. The wit is visible in the examples. The scholarship and experience are both visible in the selection of exercises.The subjects of proofs and their logical foundations have challenged the minds of some of the world’s deepest thinkers. Both the difficulties of the subject, and its beauties, are extraordinary. This book will help the reader to appreciate the beauties and overcome the difficulties.

⭐This is the most USEFUL undergrad math text I have ever seen.It covers the essentials of modern mathematics: proofs, logic, sets, number systems. I especially value the discussion of logic, sets, proof strategies, and relations.I would have preferred more discussion of boolean algebra and lattices, but that is an idiosyncrasy of mine. I also wish that it would include the killer explanation of bi-in-sur-jection, and of into/onto maps that I have sought for decades. (The best of the extant lot is in Kolmogorov & Fomin). This description should also include iso-mono-anti-tone.I am not competent to judge the freedom from mathematical error.But the exposition never seems muddled to me, and is nearly always clearer than other texts, which often grandstand when discussing deep math.

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