1. Sentential logic; 2. Quantificational logic; 3. Proofs; 4. Relations; 5. Functions; 6. Mathematical induction; 7. Number theory; 8. Infinite sets.
Helps students transition from problem solving to proving theorems, with a new chapter on number theory and over 150 new exercises.
Daniel J. Velleman is Julian H. Gibbs '46 Professor of Mathematics, Emeritus at Amherst College, and was a professor at Amherst College from 1983 to 2017. He received his B.A. from Dartmouth College in 1976, and his Ph.D. from the University of Wisconsin, Madison in 1980. His other books include Which Way Did the Bicycle Go? (with Stan Wagon and Joe Konhauser, 1996), Philosophies of Mathematics (with Alexander George, 2002), and Calculus: A Rigorous First Course (2016). Among his awards and distinctions are the Chauvenet Prize, the Paul R. Halmos-Lester R. Ford Award, the Carl B. Allendoerfer Award, and the Chandler Davis Prize for Expository Excellence. He was Editor of Dolciani Mathematical Expositions from 1999 to 2004 and the American Mathematical Monthly from 2007 to 2011.
'Not only does this book help students learn how to prove results,
it highlights why we care so much. It starts in the introduction
with some simple conjectures and gathering data, quickly disproving
the first but amassing support for the second. Will that pattern
persist? How can these observations lead us to a proof? The book is
engagingly written, and covers - in clear and great detail - many
proof techniques. There is a wealth of good exercises at various
levels. I've taught problem solving before (at The Ohio State
University and Williams College), and this book has been a great
addition to the resources I recommend to my students.' Steven J.
Miller, Williams College, Massachusetts
'This book is my go-to resource for students struggling with how to
write mathematical proofs. Beyond its plentiful examples, Velleman
clearly lays out the techniques and principles so often glossed
over in other texts.' Rafael Frongillo, University of Colorado,
Boulder
'I've been using this book religiously for the last eight years. It
builds a strong foundation in proof writing and creates the
axiomatic framework for future higher-level mathematics courses.
Even when teaching more advanced courses, I recommend students to
read chapter 3 (Proofs) since it is, in my opinion, the best
written exposition of proof writing techniques and strategies. This
third edition brings a new chapter (Number Theory), which gives the
instructor a few more topics to choose from when teaching a
fundamental course in mathematics. I will keep using it and
recommending it to everyone, professors and students alike.' Mihai
Bailesteanu, Central Connecticut State University
'Professor Velleman sets himself the difficult task of bridging the
gap between algorithmic and proof-based mathematics. By focusing on
the basic ideas, he succeeded admirably. Many similar books are
available, but none are more treasured by beginning students. In
the Third Edition, the constant pursuit of excellence is further
reinforced.' Taje Ramsamujh, Florida International University
'Proofs are central to mathematical development. They are the tools
used by mathematicians to establish and communicate their results.
The developing mathematician often learns what constitutes a proof
and how to present it by osmosis. How to Prove It aims at changing
that. It offers a systematic introduction to the development,
structuring, and presentation of logical mathematical arguments,
i.e. proofs. The approach is based on the language of first-order
logic and supported by proof techniques in the style of natural
deduction. The art of proving is exercised with naive set theory
and elementary number theory throughout the book. As such, it will
prove invaluable to first-year undergraduate students in mathematics
and computer science.' Marcelo Fiore, University of Cambridge
'Overall, this is an engagingly-written and effective book for
illuminating thinking about and building a careful foundation in
proof techniques. I could see it working in an introduction to
proof course or a course introducing discrete mathematics topics
alongside proof techniques. As a self-study guide, I could see it
working as it so well engages the reader, depending on how able
they are to navigate the cultural context in some examples.' Peter
Rowlett, LMS Newsletter
'Altogether this is an ambitious and largely very successful
introduction to the writing of good proofs, laced with many good
examples and exercises, and with a pleasantly informal style to
make the material attractive and less daunting than the length of
the book might suggest. I particularly liked the many discussions
of fallacious or incomplete proofs, and the associated challenges
to readers to untangle the errors in proofs and to decide for
themselves whether a result is true.' Peter Giblin, University of
Liverpool, The Mathematical Gazette
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