How to Prove It

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.

• 1. Sentential logic; 2. Quantificational logic; 3. Proofs; 4. Relations; 5. Functions; 6. Mathematical induction; 7. Number theory; 8. Infinite sets.

'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