Mathematical Thinking: Problem-Solving and Proofs

I ran into the first edition of this book ten years ago when taking courses at George Mason University, and really loved it. I still love it. It covers proofs from all basic "pieces" of mathematics and gives the reader a good feel for the "proofology," both in technique and fundamental nomenclature and results, that a student is expected to know when taking the first analysis and abstract algebra courses. It's not perfect though. I haven't bought the second edition, but in the first edition, Example 2.21, p.27 says: "An integer is even if and only if it is the sum of two odd integers." Obviously, it is easy to show the sum of two odd integers is even by forming the sum (2k+1) + (2l+1) = 2(k+l+1), which is twice an integer and thus even. But, if an integer is even, it can be the sum of two odd integers OR two even integers, so the statement is not complete. If small stuff like that doesn't bother you, this book is for you. The author gives solutions or hints for one-third to one half the problems depending on the chapter, which is more than enough for self-study. I would disregard the whiny one star review that is posted for this book; it is typical of someone who wants to be spoonfed mathematics.

I'm using this in an undergraduate introduction to proofs class with a focus on analysis. As a freshman, it seems a bit overwhelming at times - I wouldn't recommend it to most freshmen or even sophomores. I do feel like this does a more than adequate job preparing me for more advanced math, and goes far above and beyond similar "proofs and problem solving" style books.

This is a great book filled with great proofs and just a great amount of information. Whenever I'm unsure of a proof or need confirmation of some obscure factoid of numbers, this is where I look first. Great for building the foundations needed for learning advanced mathematics.

This is an advanced book, with a lot of information on every page. I use it as a reference book, since it has hundreds of wonderful proofs and problems, along with thorough and concise definitions for just about every major branch of mathematics. It's highly recommended for anyone who is *serious* about mathematical proofs. Although the book is packed with material, it's a small book, so it's one of the first I choose to take with me when I travel.

With a disposition towards problem solving, the book was very helpful for someone like myself who needs the mathematical tools but lacks the background. Occasionally the book lapses into the silly notion that you've seen some of the material and concepts before.