Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition) 3rd Edition

This new edition is a must have for anyone interested not only in mathematical proof, but math, logic and research writing in general. My job as CTO of a math formula software company is to present clear algorithmic ways to represent math symbols in many formats beyond LaTEX (eg e-readers). This volume takes that task a step further and shows potential authors, researchers, students and teachers how the best math articles are written today-- symbolically, logically and using natural language.Math proofs come down to the degree of rigor. Formal proofs from Euclid to Euler, and the Islamic mathematicians who moved proofs from geometry to algebra, are considered "informal" by logicians of today, who consider proofs to be inductively defined data structures, in which competing axioms can even coexist (as in Non Euclidian geometry). Today's juried math journals allow plenty of natural language that would be considered informal, and now even accept brute force, computer assisted and even some probabilistic "arguments" as a kind of proof. This wonderful volume updates the author's outstanding 2007 edition by bringing both old topics up to date with algorithms, and introducing many new topics for our algorithm driven era.There is no competing volume with this book's depth, breadth and currency, and certainly none that takes the trouble to give a tutorial on proof writing! Other older but worthy books include Discrete Math with Proof, Proof in Mathematics: An Introduction, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, and The Nuts and Bolts of Proofs, Fourth Edition: An Introduction to Mathematical Proofs. In the more general field of research writing, The Craft of Research, Third Edition (Chicago Guides to Writing, Editing, and Publishing) and Critical Thinking, Reading, and Writing: A Brief Guide to Argument are must read classics for those publishing research and articles.Since proofs take us into advanced math areas such as analysis, group theory, ring theory and number theory, at least Calc 1 is assumed, Calc 2 being better. ODE's and PDE's are not required, but would be a plus as the authors do not shy away from applications even in a proof oriented volume. What about Library Picks four standards for reviews (Scan/refer/read/study)? I'm giving this max stars for all four, whether you are into self study or are teaching/taking a course, or getting the volume for reference.The authors' website has been updated for this edition, and thus researching encyclopedic keywords, as well as self study, are outstanding. The pedagogy also is sound for a one or two semester course dovetailing calc 1, 2 or 3 into more formal areas for math majors, engineering grad students, etc.If their outstanding chapter on how to write proofs for publication isn't enough, Reading, Writing, and Proving: A Closer Look at Mathematics (Undergraduate Texts in Mathematics) is the current best of breed in weaving "how to do proofs" with "how to write them" for juried publication. Combining Cupellari and Daepp is an outstanding (albeit expensive) strategy to accomplish this combined learning - teaching agenda of both doing and writing outstanding proofs, whether you're a student or researcher. With less of an investment, you get a slice of both here in this volume.On a personal note, our lives are executions of our own internal proofs, and most experiential logic is statistical--we execute those routines that seem to work best on average or in sum from our lives. Studying more formal proof structures (Direct, inductive, transposition, construction, contradiction, etc.) can really enhance our ability to make better choices. In addition, seeing the beauty of certain math proofs (check out Proofs from THE BOOK) can be inspiring and fun, just like an unexpectedly brilliant chess move. Euler was renowned for this, even in topics that seem "dry" to non mathematicians, like infinite series!

Although the price is a little high, the book is well worth the price. This is one of those books where the authors leave nothing out by assuming you know something that you don't, but yet are concise in their delivery, without being wordy and adding a lot of unneeded discussion. Part of the learning and discovery are in the well thought out exercises, so you'll want to do all of them. You shouldn't have any problems doing them all, because there was nothing left out in the chapters which would not allow you to complete all the exercises.The book covers all the basics of sets and logic, in case you are not up to speed on those topics. Later chapters focus on various topics, so you could skip some of those that don't interest you. This book is highly recommended before undertaking a rigorous calculus or other higher level math course, since the study of proofs seems to be something that is left out these days in high school and college.

Really good for mathematical proofs. Needs a little more explanation about set theory and how x and y can be moved across sets A and B because it seems vague. Also, needs to clearly state that if any number x is even, then x = 2k where k is an integer. Also if any number y is odd, then y = 2k + 1 where k is an integer. That wasn't really as clearly explained as I would have liked it to be.

The biggest weakness with the text is explaining intermediate steps within each topic. I often need examples to increase in difficulty at a slower rate, or for a little more explanation over why each logical step is what it is This is particularly problematic because the whole idea of proofs is an orderly explanation of logic.

This is the best proof book by far.The other books I tried areMathematical Thinking: Problem-Solving and Proofs (2nd Edition)How to Prove It: A Structured Approach, 2nd EditionThis book is much better than the other two book.The nice thing about the book is that the chapter is organized by method of proof (direct, contradiction, induction, ...).This really helps to improve each proof method, instead of using only one method you are familiar over and over.

I needed this book quickly for a summer class. It arrived quickly and in perfect condition (exactly as described by the seller). I will definitely keep this long past the end of this course.It is a great introduction to logic mathematical proofs. It is easy to read and follow, and I'm sure will be invaluable to me as I progress through my academic career.

Sometimes when you dot all the i's and cross all the t's you lose the big picture of what is going on. I found this being done in a number of places. Here is an example, Theorem 11.13, Euclid's Lemma on page 276. This is an important result used for proving the Fundamental Theorem of Arithmetic. First I will show the proof as given in the book and then the more compact and easier form in which it is usually given.Verbatim from the book: Let a, b and c be integers where a not equal to 0. If a | bc and gcd(a,b) = 1 then a | c.Since a | bc, there is some integer q such that bc = aq. Since a and b are relatively prime, there exist integers s and t such that as + bt =1. Thus,c = c x 1 = c(as + bt) = a(cs) + bc(t) = a(cs) + (aq)t = a(cs + qt). Since (cs + qt) is an integer, a | c.Here is a much more compact way of saying the same thing that is much easier to follow.Since a and b are relatively prime, we can find integers s and t such that as + bt =1. Multiplying both sides by c, c(as) + c(bt) = c x 1 = c. a(cs) + (bc) t = c. Since a divides the first term and a divides bc, the left side is divisible by a and therefore the right side, c, is divisible by a.