How to Prove It: A Structured Approach, 2nd Edition 2nd Edition by Daniel J. Velleman (PDF)

Description

Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches 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. To help students 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. Previous Edition Hb (1994) 0-521-44116-1 Previous Edition Pb (1994) 0-521-44663-5

User’s Reviews

Editorial Reviews: Review “The prose is clear and cogent … the exercises are plentiful and are pitched at the right level…. I recommend this book very highly!” MAA Reviews”The book provides a valuable introduction to the nuts and bolts of mathematical proofs in general.” SIAM Review”This is a good book, and an exceptionally good mathematics book. Thorough and clear explanations, examples, and (especially) exercised with complete solutions all contribute to make this an excellent choice for teaching yourself, or a class, about writing proofs.” Brent Smith, SIGACT News Book Description This new edition of Daniel J. Velleman’s successful textbook contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. About the Author Daniel J. Velleman received his BA at Dartmouth College in 1976 summa cum laude, the highest distinction in mathematics. He received his PhD from the University of Wisconsin, Madison, in 1980 and was an instructor at the University of Texas, Austin, from 1980 to 1983. His other books include Which Way Did the Bicycle Go? (with Stan Wagon and Joe Konhauser, 1996) and Philosophies of Mathematics (with Alexander George, 2002). Among his awards and distinctions are the Lester R. Ford Award for the paper ‘Versatile Coins’ (with Istvan Szalkai, 1994), and the Carl B. Allendoerfer Award for the paper ‘Permutations and Combination Locks’ (with Greg Call, 1996). He has been a member of the editorial board for American Mathematical Monthly since 1997 and was Editor of Dolciani Mathematical Expositions from 1999 to 2004. He published papers in the Journal of Symbolic Logic, Annals of Pure and Applied Logic, Transactions of the American Mathematical Society, Proceedings of the American Mathematical Society, American Mathematical Monthly, the Mathematics Magazine, the Mathematical Intelligencer, the Philosophical Review, and the American Journal of Physics. Read more

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

⭐Before buying this book, I struggled in math. I excelled at “calculating” stuff by simply plugging in numbers into some sort of equation our high school teachers would spoil us with, but when I got to college, I had to start thinking abstractly- and it bothered me a lot, because I had no idea how to test or prove the logic of some statement. I was doing very poorly in linear algebra and desperately needed help- lo and behold, my professors weren’t helpful (at all). Someone recommended this proof writing book to me, and I am VERY grateful for that referral.The book takes the average student (it’s shocking with how little math background one needs) and introduces him to basic boolean logic. You know, material like “If A is true, and B is false, then A implies B is false.” In a discrete mathematics course, one would call this “truth tables.” From there, the author takes the reader into set theory, basic proofs, group theory, etc- and into more advanced topics, like the Cantor-Schroeder-Bernstein theorem, countability, etc. So what makes this book stand out?(1) Readability. Many math professors stop just short of taking pride in how confusing, abstract, or daunting their lectures can be. Velleman, however, goes the extra mile in the text to see that the reader UNDERSTANDS the logical buildup and concepts of mathematical proofs. Sure, set theory can be confusing- but after reading several other texts in discrete math, including “Discrete Math and its Applications” by Kenneth Rosen (if you’re reading this, no offense) I’ve found that Velleman by far writes the most comprehensive and cohesive explanations for understanding set theory. Making the material accessible is the mark of a real “teacher,” and if you read through this book yourself, I believe you’d agree that Velleman is a pretty legit teacher.(2) Examples. There are plenty- plenty that Velleman works out himself. Reading the examples alone- and actually taking the time to understand them- is a task that’s up to the reader, obviously, but they do show results almost immediately in understanding discrete math.(3) Problems (exercises). There’s never a shortage of exercises, I found, as I tried to work through the problem set. There are plenty. Fortunately, there are some answers in the back, but just enough so that you can verify to see if you’re understanding the material, and not enough so that you find yourself copying every answer in the back (even the best students get tempted to do that). Velleman gives the proper amount of answers in the back and a ton of exercises to do. If you complete them all properly, you’d be far ahead of the curve amongst math majors.I know my review may have been too wordy, or too optimistic. However, my feelings are very honest and not exaggerated: this book is written so one can learn discrete mathematics, and really helps the reader understand what higher math is all about- and how mathematicians think, write, and communicate. This book deserves an A+, and I’ve only given that score out to a handful of books.

⭐A fine book. It is very clear and concise, and in my opinion very enjoyable. It goes over the foundations needed to writing proofs (basic set theory and logic) and provides a lot of exercises and examples. The first chapters deal with the ideas of logic and sets and of the different ways of proving different types of statements. In the later chapters, extremely important math concepts that are not essential to proof writing are introduced – relations, functions, and infinite sets – providing more training in proof writing. The concept of induction (both ordinary and strong) is also introduced in the later chapters.The sections are mainly very clear and concise explanations of the concepts, together with examples, theorems, and definitions. Velleman is a fine proof writer; his proofs are very readable and it is very easy to understand them. Therefore it is very worthwhile to study them and perhaps to even try to mimic them, to some extent. The end of every section is a very large set of exercises. Some exercises have solutions in the back of the book, but beware that for most of them, no solution is provided. This is a great drawback, in my opinion, and I wonder why Velleman decided to leave so many exercises without a solution. However, the exercises are very good! The first exercises in every set are generally quite easy, and the last ones can be quite difficult. Many of them are very interesting.Here’s a very interesting exercise from the section dealing with strong induction:”The martian monetary system uses colored beads instead of coins. A blue bead is worth 3 Martian credits, and a red bead is worth 7 Martian credits. Thus, three blue beads are worth 9 credits, and a blue and red bead together are worth 10 credits, but no combination of blue and red beads is worth 11 credits. Prove that for all natural numbers n greater than or equal to 12, there is some combination of blue and red beads that is worth n credits.”

⭐I hesitated a little in buying this book, because even a beat-up used copy is pretty expensive, and a new copy is sold at the prevailing textbook ripoff level. But the book is really worth what I paid for a highly used copy. Why? Because the author, instead of merely wanting to show off his erudition, is genuinely interested in teaching math, and is genuinely good at it.This is what I once heard described as a “talky” book. There is plenty of explanation and little if anything is left to chance. Dr. Velleman works things through step by step, with clear explanations and enough examples to be sure the topic can be easily understood.The book is typical in content for a proofs course: there’s the introductory chapters on logic and set theory, and a graduated approach to formal proofs. Major proof methods are covered in detail. If you work your way through this book, you’ll be very well prepared for more advanced math courses such as abstract algebra, advanced set theory, etc.The book is, in fact, eminently suited to self-study, very much more so than many others on the same topic. Advanced students just looking for a quick review on their own may find all the “talky” stuff annoying, but the beginning self-student will really appreciate it.The problems are ample and solutions provided to some of them. They are sufficiently interesting to help teach the material, but generally not super challenging, at least in the early chapters. I think that’s good; reinforcing the basics is really important and if you want a greater challenge, other books or online courseware can provide that.Bottom line: an excellent text for the beginning proofs student, and especially good for self-study.

⭐* PhysicalThis book has 384 pages printed in B&W printed on good quality paper.* Target audience, A-level / H.N.D. Undergraduate / Graduate / Postgraduate?In my case this book would be helpful to those studying (some) H.N.D and through to selective undergraduate computer science and mathematics in whatever form you need help with. The major bits i feel could help are the logic parts to extend your grasp of these theoretical topics and concepts.* What’s the best bits then?To be honest, i have previously little grasp of proofs except in niche topics as required for a course not wholly based on this topic. Say studying ‘Mathematical Induction’. If you consider this describes your experiences then i feel this book is to be a great help. This book creates a better grasp on proofs NOT jumping feet first into the MUCH deeper parts. The bits i felt the most benefit from is the ‘Sentential Logic’, ‘Quantificational Logic’ and ‘Proofs’, ‘Relationships’. The later parts of ‘Functions’ i have already covered elsewhere in more depth in other works and courses.The parts i engaged with and covered for the first time in my experience being the mathematical symbols for connectives ‘^’ , ‘v’ ‘¬’, ‘<->‘, ‘->’, ‘<-', member of sets, not member of sets and quantifiers and few others. The depth in which it is expressed i felt carried REALLY well and engaged my thoughts and pushed me to grasp stuff i had seen in other manners but not in formal set notation before. These topics are to help in other areas and ricochet into other algebraic courses you may reacquaint yourself with.The depth i feel has been curtailed to make a foot - in- the - door and not to cover every proof that's possible in mathematical terms. There are many questions and answers to help clarify issues if you're confused with too.* SummaryThe author knows how to put an book together to help the student new to some of these areas in a way that's really helpful and enjoyable to encourage the study of this topic further down the line. I really enjoyed this book and found it useful to tidy up a few areas i could not make progress with. The mathematical explanations of qualifiers, 'and', 'or', 'not' symbolic notation is a nice touch and i wish other writers would explain these before going into depth. ⭐This is a beautiful introduction to proofs, propositional logic, sets, relations and functions for first year university students. It is clear that Velleman's first and only interest is to keep the reader engaged and make sure that the concepts become accessible and sink in. His style is vibrant and engaging, and I actually found myself looking forward to reading this while I was studying for part of a Computer Science exam.That said, this book truly comes into its own if you read it cover to cover, or at the very least chapter to chapter. This is not strictly speaking a reference text, and Velleman places more emphasis on fluidity and clarity than on organising the material into neat compartments. The chapters aren't unreasonably long, however, and with a bit of dedication this book will take you a long way. ⭐I struggled severely in my logic/discrete math course. I read a few good reviews on this book on amazon.com, and bought it. It's such an easy read, while being thorough at the same time. Everything is explained perfectly and I couldn't ask for a better book. This is one of the few books I feel should be in any computer science student's bookshelf. ⭐Great book. ⭐Fantastic book...really good if you like to use your brain.

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