Customer Reviews

Proof, Logic, and Conjecture: The Mathematician's Toolbox

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1 star:		(3)

 

 

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.

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......

The book covers a wide variety of topics that are important for a student in mathematics; however, there are way to many errors and confusing phrases/explanations throughout the text (i.e. pg. 100 example 13). Another section of the book that might need some work is the exercises at the end of each chapter. It seem that in many cases the text does not make bridge like transitions to the problems at the end of each section (i.e. PG 162 in section 5.3 has a problem (#15) that should be in section 4.5). Along with several spelling errors and/or improper grammar, that part is easy to spot, there are some errors in the "Solutions and Hints to Selected Exercises" section, not to mention that this part of the text is not all that helpful unless you are looking for additional confusion. By that I mean a student starting higher mathematics will probably struggle with it. The book needs to calm down if it is going to be a textbook....

Requires, but rewards, close attention and honest effort. Comprehensive and well-organized. If you find this book beguiling and you want to survey a wide variety of topics that you might also enjoy, try Kasner and Newman's "Mathematics and the Imagination."

Chapter 1 seemed to have nothing to do with the rest of the text. Playing Nim and being told the scientific, legal and philosphical ways to prove things had nothing whatsoever with learning how to prove things in mathematics.

The truth tables were helpful, but the "or" table needed more explanation from the beginning.

2.1 ex 5d: it's confusing to students to have "trick questions" when learning a concept for the first time. "Tricks" should either be saved for more advanced students or not used at all. The many incorrect statements, proofs and such throughout the book violate the "never write down the wrong way to do something rule". It was extrememly confusing. The defininitions for converse etc. were clear until ex2 p. 29 where I got completely lost. Tautologies were helpful.

E.S., E.G., U.S., and U.G. were only muddying up the waters and since other texts don't use them, they seemed pointless. Ways to use/prove tables were helpful. Induction was very confusing to start with, especially when using summations, but then seemed to clear itself up in later sections.

Get rid of Naive Set Theory. It's wrong and beginning set theory students are only misled by being shown the wrong way first! The union of the power set section needs a clear example as to what the union of a power set is!

I'm not sure that beginning students are ready to critique quite so many proofs, especially the "borderline" ones.

Figure 7.11 is completely lost any effect on me. I don't understand it at all.

A glossary in the back of the text would also be useful.

This book really sucks. One of the worst books I've got in my collection of math books