14 of 14 people found the following review helpful:
4.0 out of 5 stars
All you need and more, October 30, 2002
This review is from: Schaum's Outline of Set Theory and Related Topics (Paperback)
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.
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7 of 7 people found the following review helpful:
5.0 out of 5 stars
The Front Door to Abstract Mathematics, December 1, 2009
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.
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