Customer Reviews

Schaum's Outline of Set Theory and Related Topics

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This is THE book to learn set theory if you are not a mathematician or mathematics major. Set theory is presented in a non-axiomatic manner. All the basics are covered: elementary set theory, cardinal and ordinal numbers, axiom of choice, and a brief intro to logic. The number of examples is huge. If you need examples when learning, then this is the book for you!

Like the other books of the Schaum's Outline series, "Theory and Problems of Set Theory and Related Topics" provides you with hundreds of problems, which have been worked out step by step, to show you how the theory works. This is similar to a professor in college showing you the problems on the board, but these don't get erased and are much more legible.

In addition to the solved problems, there are supplementary problems to test your understanding of the material. At the end of each chapter, the answers are printed for you to check your work.

The discussions of set theory include definitions to ensure that you are familiar with the lingo. There are also plenty of examples to illustrate the author's meaning. With all the information on set theory, there are also chapters on quantifiers, boolean algebra, logical reasoning, and the algebra of propositions.

This book provides some good assistance in learning set theory.

This book has been in print for over 45 years, and there's a good reason for it. It's excellent.

Whether you are a math student or a student in another discipline struggling with the mathematics necessary in your field (anything beyond calculus), you should read this book. Immediately.

Set Theory is the foundation of modern mathematics, and if you don't understand the basics of it you simply don't understand math. At all. Really.

This makes it all the more amazing that it is possible to get a PhD in mathematics without ever taking an actual class in set theory, and during my education I found that even professional mathematicians would sometimes make elementary mistakes with set theory.

Don't let this happen to you! By this book. Study it. Carefully. If you are a math student, it will strengthen you. If you are a student from another discipline, you will likely find that many of your previous mathematical difficulties begin to melt away.

I have seen many books on set theory over the years, and I have never seen one that I would recommend before this one. Never. In fact, for anyone interested in or forced to work with higher mathematics, I'd recommend this as the very first book they read.

However, for anyone reading this book, there is an important error which I will discuss because you may find it helpful. The definition of an ordered pair (a,b) is given as {{a},{a,b}}. You are told that with this definition (a,b)=(c,d) if and only if a=c and b=d.

But if you try you will find that you are not able to prove this. Indeed, set theory as developed in this book is insufficient to allow you to prove this result. In more advanced treatments of set theory, you will find out that this result relies on the Axiom of Foundation. Or you can just take the author's word for it, which is the only sensible course of action at this point.

The issue is that set theory becomes a deeply technical subject, full of subtle distinctions and formal logic. Fortunately, it isn't necessary to know it that well unless you really become interested in set theory for its own sake, and after reading this book you just might! The author is mostly very clear about the issues with this development of set theory. For example, there is a section completely devoted to showing you a variety of ways in which set theory as developed in this book is logically inconsistent. And this is exactly how things happened historically: after mathematics was set up on the basis of set theory, it was found that set theory itself -- as they understood it at the time -- was deeply flawed. This necessitated a careful, difficult, axiomatic overhaul of the subject which is of little practical value to people who actually just want to USE set theory to help them with other math, like engineers, physicists, or even most mathematicians. And those are the people this book is aimed at.

That said, the omission of the Axiom of Foundation from this treatment is perfectly understandable. However, it would have been nice if the author at least informed the reader that the set theoretic definition of ordered pair given requires additional set theory to completely justify.

So this book is not perfect, but in spite of its flaws it is an outstanding book, and after a short discussion of the ordered pair business I have recommended it to many people over the years. And now I am recommending it to you. Reading it will give you a new found power when dealing with abstract mathematics no matter who you are, and the mathematical prerequisites for this book are essentially non-existent. Set theory is where abstract mathematics starts, and this is the best first book on the subject I have ever seen.

I went through the first chapter and worked all of the problems.

First, the typos:

On Page 10: About half way down it says "(a) on list or B"; it should be "(a) on list A or B"
On Page 33: The answer for 1.68 says "a is an element of X or a is not a subset of X." After the or it should be "not an element of X."

Second, the incorrect answers:
1.61: (b) should be 65 and (c) should be 25
1.63: They drew the Venn diagram correctly but then got (d) and (e) wrong. (d) should be 8 and (e) should be 12.

Third, the incorrect problems:
1.74: The right side formula is incorrect, I don't know what the correct formula should be.
1.75: The right side should be n/(2n+1) not 1/(2n+1). I checked this with n = (1,2,3) and the proof by induction worked with n/(2n+1).

If someone points our where I am wrong I will change or delete my review. As it is, six mistakes in one chapter is a a lot in my estimation, especially when the problems are incorrect. The whole point of Schaum's is to provide lots of example problems. If lots of the sample problems have mistakes it really cuts down on the usefulness of the book.

This is an excellent introductory text on set theory. It provides a fast and thorough coverage of the basics in an economical, if terse, style. If you diligently go through this short 200 page book you will have a solid grasp of set theory that you can immediately use on the job.

I am an IT consultant so, although this book has a much wider audience, I am focusing my review on how this book applies to IT. Because many e-commerce systems are being designed using role-based access controls that are independent of any specific application it is essential that the designers have a thorough understanding of set theory. As such, I think this book is an indispensable quick reference guide for IT security specialists, system architects and QA team members who need to brush op on the subject. These folks design, validate and verify systems that require strong security from within and outside of their organizations. This type of work requires a provably correct design that can only be approached with set theory. Moreover, the testing of the design, as implemented, also requires an understanding of this subject in order to develop a viable test strategy and produce test cases. Given the fact that most IT professionals are not versed in set theory, this book is one of the quickest ways to get up-to-speed.

Among the highlights of this book are: the paradoxes in set theory, which is "must reading" regardless of how you intend to use the knowledge and skills that this book provides, algebra of propositions and logical reasoning.

This book is clearly written, however, the material requires a strong dedication to learning because the subject is complex and dry. No author, regardless of his or her writing skills can change that. If you need to understand set theory and don't know where to begin, this book is one of the best starting points I have come across.

The material is well presented with numerous problems that drive home the concepts in the book. The reader must be a little careful as there a few errors in the Problem Solutions though. Overall, this is an excellent text.

This is a useful book. Goes quite deep down to the basics. I am not a mathematician but a linguist and needed to know more of set theory of our language and logic class and this book helped me a lot. Good reference book for non-mathematician interested to learn more about sets and functions!!

Like all books in the Schaum Outline series, it you want to get up to speed quickly in a particular field, without have the burden of the author assuming you have foundational knowledge in the topic, then you will love this book.

The best feature about these books is the examples and exercises frame what you have read and learnt very well so you find that they greatly helping in cementing what you have read - often an example is worth a thousand words. Schaum, really do this very well.

First class and very well written and logically presented.

I used this book in a Mathematical Logic course, and I found it helpful when giving examples for Set Theory. It was definitely easy to follow, but we didn't really reference it that often. However, I wouldn't look at it on it's own, but then again it says it's an outline so I doubt one would mistake it for a text book. It's a decent guide to set theory, especially for people who aren't that into math. There are better books out there for people who want to go further.

I thought this book was a great guide. It is a comfortable border between a refrence book and a self-teaching guide.