Introduction to Mathematical Structures and Proofs (Undergraduate Texts in Mathematics)

This is a delightful book. It is well written, clear and even funny in places. Dr. Gerstein has done an excellent job of making his subject appealing. The only flaws I can find are a few misprints of Teeorem numbers when referring to topics covered elsewhere in the text, an error certainly not worth the subtraction of even half a star.Dr. Gersten begins by discussing logic, then set theory. He then teaches topics in Functions, finite and infinite sets, combinatorics and number theory as a way to teach the art of writing clear proofs. Toward the last half of the book, there is more emphasis on the mathematics than teaching how to write proofs, but the real purpose is to lead the reader onto the path to becoming a mathematician. He succeeds beautifully.The competition for the undergraduate’s purchase would probably be Hammack’s Book of Proof, another excellent text. Hammack takes a different approach, using the construction of proofs as a way to introduce the student to mathematical concepts. This book is briefer and written at a more basic level than Gerstein’s book. If you are self-assured in math, proceed directly to Gersteins book. However, why deny yourself the pleasure of reading either of these books? They give different looks at their subjects, and I doubt that many will regret buying and reading both books, starting with Hammack. I certainly don’t.

This book had a huge amount of material covering a wide range of topics. The treatments of the diverse areas are well integrated. The exercises are difficult but manageable based on the discussion. I would recommend it to others who, like me, have a background in an application of math (physics in my case) but have never surveyed math rigorously.Few or no errors found despite painstaking study.The chapter on number theory seemed overly ambitious and I had to skip over the last part of it.

It's a good book for a hard subject, it's a very readable textbook and the author's wit adds a personality to the text. The only fault is that there aren't a lot of practice problems for some sections, and only the odd answers have solutions in the back of the book.

We are using this book in my Foundations of Math class, which deals a lot with graph and set theory, as well as proofs. The professor chose the book after listening to the author speak at a conference, and I'd say it was a good choice. The book reads very easily, and the problems do well at not being overly complex, while still being challenging enough to conduce learning.

That’s all - 5 stars

Dr. Gerstein is now a friend of mine. I highly recommend this textbook in the Field of Logic and Mathematics.

This is a pretty good introductory book for students new to proof-based mathematics, but there are better options out there if you are new to proofs and/or self-studying. I would probably give this 3.5 stars but rounded up because the content really is quite good. However, I *do not* think this book is suitable for "learning proofs" in and of itself.The very first example in the book is a very interesting example of an abstract axiom system, with a single axiom and two logic rules. While it is just a toy example, and may not add much to the text, it made me think differently than other proof-learning books I've read. (I am not sure if this kind of thing is standard in mathematical logic courses/texts.)Most of the content in Chapters 1-4 are also presented in other texts. For example, a lot of this material is in Velleman's well-known text "How to Prove It", but I actually prefer this book to that one. The author here clearly put a lot of effort in making these concepts explainable.I was most interested in the long chapter on Combinatorics (Chapter 5), which is relevant to my interests (probability & statistics). While the content is very good, and he explains the concepts well, the chapter did seem to drag on and I think could have been improved if the abstract material was better connected toapplications (e.g., discrete math or probability).Some of the drawbacks/complaints I have include:1) The major disadvantage, especially for someone using this for self-study, is that the book includes solutions for only approximately half the exercises (and in multi-part questions, only up to half of those). Furthermore, nearly all of the included solutions are incomplete, hints, or just outlines. For such a "transition" course (i.e., a first proof-based course), people such as myself will struggle when you're not sure if you've taken the correct approach.2) Some of the more important concepts (e.g., theorems or corollaries) are presented in the exercises. And, as mentioned above, these exercises might not have (full) solutions. So in a few cases in which the exercise asks you to "prove or disprove", students who can't complete the exercise won't know whether it is a correct theorem.3) There are some very minor typos throughout, although from what I've seen so far they do not impede understanding. For example, when referring back to an Example or Definition, the number might be off by one. There are also some spelling typos here and there.4) One very minor complaint I have is the size of the book. My hardcover copy is quite a bit larger than my other Springer *Undergraduate Texts in Mathematics* books. As a result, the font appears larger than it needs to be. However, this didn't bother me enough to give the text a lower rating, but perhaps others would be interested to know this if they prefer uniformity.Overall, I liked this book and I think it has helped improved my understanding of basic abstract math concepts. I would still recommend other texts over this one, but it would not be a bad addition. If you want to actually learn proof methods, my first recommendation would be Solow's How to Read and Do Proofs for beginners and "How to Prove It" otherwise. Another really excellent choice would be Reading, Writing, and Proving by Daepp and Gorkin; there are detailed solutions to the problems in this one, and (so far) it seems much better-suited for beginners or those who are doing self study.

Un libro excepcional, tanto por los ejemplos elegidos para introducir los conceptos fundamentales, como por el mantenimiento del rigor, aunque exento de una excesiva aridez formal. Muy recomendable para lectores interesados en adquirir una buena base para poder abordar, con la preparación suficiente, temas más complejos de análisis matemático.