Customer Reviews

A Transition to Advanced Mathematics

5 star:		(10)
4 star:		(11)
3 star:		(2)
2 star:		(1)
1 star:		(13)

 

 

I've honestly never understood the negativity thrown towards this book.

This book introduces basic proof-writing techniques, naive set theory, cardinality, functions, and also includes a very brief introduction to abstract algebra and real analysis. I thought that the section on proofs was quite well done even though there are better textbooks which cover this topic more extensively with more examples. The examples are generally simple and are pulled almost exclusively from number theory, but they provide understanding. The set theory chapter is not very rigorous, but it meets the goals of the type of student who would take such a class. A more rigorous (i.e. axiomatic) introduction to set theory would require an entire semester, and the goal of this book is simply to provide students with just enough tools to prepare themselves for classes like Abstract/Modern Algebra, Real Analysis, and Set Theory, but not much more. Also, I consider the proofs to be generally well-written although there are occasional mistakes.

The chapters introducing abstract algebra and real analysis don't provide much information, but only serve to give students a taste of the fields. I felt that they should have either been omitted (since you won't learn much) or they should have been introduced with more rigor and explanation.

Some people have complained that the exercises were too hard. I felt that the opposite was true. All of the problems are either trivial or of medium difficulty. I think that the problem was that the proofs presented in the text were even easier. The proofs are generally well-written, but it would have been nice to see more challenging proofs to give the student a better idea of how to work the more challenging homework problems.

It's not the best or the worst text that you can find on this subject. At minimum, it's at least underrated (though overpriced).

Firstly, I must clarify: other reviewers have said that the book claims 1 + 2 does not equal 3. They are not reading carefully enough. I have the 5th edition, and this is exactly what it says (on page 18 for the 5th edition).

"The sentence 'x1 is equal to x2 + x3' is an open sentence with three variables. If we denote this sentence by P(x1,x2,x3), then P(7,3,4) is true since 7 = 3 + 4, while P(1,2,3) is false."

Now to translate for them, in the case of P(1,2,3)
x1 = 1, x2 = 2 and x3 = 3. Thus the sentence says 1 = 2 + 3 which is false, not 1 + 2 = 3 as other reviewers have claimed. What is confusing is the order and the abstract expression. In the order given P(3,1,2) would say 3 = 1 + 2.

Now as for my rating, I do not claim that the book is ideal. But, it does fill a gap in math education, and there are not many other books like it. A good replacement would be an introduction to logic, or to set theory, but this book contains both. I hope we will see more books like this in the future, and if we do then we may eventually find one worth bragging about.

Currently, though, there is no such book that I know of, and believe me I have seen a lot of books. In the meantime this book can serve as a valuable reference. So, as per my title: read carefully and think! If you do that, then the contents of this book should be crystal clear to you.

We've been using this in our Introduction to Mathematical Proofs class (an upper-level undergraduate math class). I was a little nervous after seeing so many bad reviews here, but it's been a fine textbook. It's easy to learn the material from the book, and the question/problem sets are useful too. I don't have any other books of this type to compare it to, of course, but the layout of the text and material is clear, with definitions and properties given as needed.

Most of the problems of the next edition are in this but in a different order. Its just a money grab

Smith , Eggen, and Andre do an excellent job presenting the basics of logic, set theory, group theory, analysis, and other high level math to a novice crowd. I'd highly recommend this book (having used it as a student) for a post-calculus sequence transition class.

The heart of the book is exercises in grading proofs. These are very useful - especially to future teachers.

There is a lot of overlap between the first four chapters of this book and what is often taught in discrete math but the book is more thorough and rigorous. In addition, there is a chapter on cardinality, and introductions to abstract algebra and real analysis.

Teachers may not get a chance to take many upper level undergraduate courses that are oriented to math majors. This book is a transition to math for mathematicians (rather than engineers and science students) and covers the fundamentals in a very accessible manner.

This book clearly does NOT suppose "number theory"---unless you mean that a even number plus an even number is even!
The point of this book is to help students make the transition from plug and chug calculus, linear algebra, etc, to thinking through WHY theorems are true, and how to go about proving them. No, the material is not advanced. The focus is on the language and techniques of proofs. Who wants to struggle proving the fundamental theorem of algebra without even knowing what induction is?
I've taught with this text for several years now and find it presentation and coverage, as do the students. Yes, the cost is outrageous...as are most textbooks. Buy it used.

I just began using the seventh edition of this book and before I even got into the actual chapters (in the Preface for Students) I noticed an error. On page xv "For all x in Z[the set of integers], x+0=0,x*0=0..." I don't have my PhD so I could be wrong but I do not think x+0=0.

I will reevaluate this textbook at the end of the semester but this was not a good start.

Let me begin by addressing the negative comments:

I see many complaints about the cost of the text. Let me explain a bit how the process works. In math, when one submits a textbook for publication, especially if it is your first textbook, you are quite happy to simply get your book published. When I had the course 11 years ago, using a previous version, the cost of the text was much lower. Often, if the publisher is small like this one is, they do not have any control on cutting the cost of their texts and the author, feeling loyalty to the small publisher who took a chance with their text, does not wish to change publisher. Blaming the authors for the cost of the textbook shows a real naivety as to how the process works. I invite students to research costs of other textbooks in math from courses they had several years ago. You may encounter several shocks.

Secondly, as to the actual content of the textbook. I see many complaints that the textbook doesn't explain things well enough from students currently taking the course. I feel that amazon should ban math students from commenting on a text that they are currently using! In my 10 years of teaching, I have never seen a mid-low level math student ever say anything nice about a textbook they are currently using. As to specific complaints about the textbook, when I had the course taught from this textbook about 11 years ago, I had a similar complaints as one expressed here: namely that the book ``does not show you how to do proofs.'' This is clearly written by students who do not know how to do proofs! There is no ``way'' of ``doing proofs.'' It is an art which simply put, requires practice. This textbook is one of the best textbooks at isolating problems which allow students to practice.

Thirdly, if the publisher of a textbook does not actively review a book there will be typos, period. This is something that I have experienced in writing nearly 600 pages of text. I challenge every person who wrote a bad review about the typos to take as much time as they'd like to write up a 2 page proof of something, anything they choose. They can take as much time as they'd like with the only rule being that they are the only ones who are allowed to read the book. I promise you, I guarantee you, that there will be typos in your work. How do I know this? Because this is an assignment I give every semester in my proofs writing course! I have NEVER had a student turn in such a paper without several typos. Moreover, I have not found a large number of typos in this book myself! Again, I believe students complaining about the typos in this text have little experience reading math textbooks. The Errata for any LMS textbook is several pages long!

I didn't see any reviews in which a student says ``I don't like this book, and this other book 'blank' is better because 'blank'.'' Perhaps before writing a review you ask yourself whether or not

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Now let me get the pros of this textbook. Let me start off by saying that this textbook is the reason why I changed my major to mathematics, and subsequently went on to get my PhD in math. This is no exaggeration. The textbook starts with basic logic, moves onto set theory, and function theory. It goes onto cardinality arguments, group theory and real analysis. In short it covers every topic one encounters in their mathematical undergraduate experience from the point of view of somebody just starting out in advanced mathematics. This is the only textbook I know which tackles this. The first few sections (before the cardinality chapter) are, for starters, done in the obvious order. This seems silly to point out, but having looked at many other texts that ``teach proofs'' this is one of the few that actually manages to do this correctly.

Another bonus is that it doesn't obsess with induction. In my experience, from teaching the higher level courses which are built on this one, every student can do basic induction problems. Spending too much time on induction is like learning how to hit a baseball by learning how to plant your foot for hours a day (yes, the Karate Kid is fiction!) This text, unlike others, introduces the student to the real ``players'' in advanced mathematics that they have not seen before: equivalence relations and functions (in a formal sense). These two concepts are both put under the heading of ``relations.'' It also convinces the reader that they have been studying the former, equivalence relations, throughout their mathematical career!

The textbook also does a wonderful job of balancing what belongs in such a foundations course without skipping any of the basics or going into things which could best be classified and understood inside of the philosophy department. Again, there is no way that a student taking the course would know this! This is because they haven't stuck around long enough in the subject to learn whether or not what they learned from the text actually has any benifit to them in the real world. With a whole chapter dedicated to it, I feel as if though they do not achieve the wonderful balance they get in the other areas of the textbook: between too much and too little. The study of cardinality theory can either be taught in a very abstract way (bordering again on phil) or in a very concrete way. I feel as if the authors attempted to get in the middle, which is nearly an impossible task for this particular area. I don't feel like they did a bad job here; I cannot see a better way of doing what they attempted to do, but I do feel like they attempted an impossible task and failed.

My teacher selected this book for the Set Theory class that I am taking. This book has a lot of information in it and some great examples. The material is tough but if you work through the exercises at the end of each section you will have a good handle on the material. This book, unlike many other math books, is actually fairly simple to comprehend as well. All in all its been a great book.