Student review: How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

May 27, 2010

Author: Daniel Solow

Edition: Fifth revised

Publisher: Wiley and Sons

Pages: 301

Price: £49.50

ISBN 9780470392164

The transition to university-level mathematics from school can be quite daunting, as the subject appears to take on a whole new perspective: that of the mathematical proof. Although proofs form the fundamental building blocks of the subject, they are often neglected in pre-university education, leaving undergraduates facing what appears to be a foreign language. It is this predicament that the author of this textbook seeks to address.

The author's approach is to form a structured framework of key proof techniques, starting by explaining the key terminology before beginning with the simplest understandings of logical concepts and truth tables, and then slowly moving on to other methods such as proof by contradiction and induction. The key to writing many proofs involves a certain amount of creativity, intuition and experience, which cannot easily be taught in one textbook. Indeed, Solow acknowledges this, stating that the techniques of this book are intended only as a guide to getting started with proofs, and he describes how, when and, most importantly, why the different proof techniques are used.

The proofs used will make this book immediately accessible to the new undergraduate. It starts with simple ideas and familiar results such as Pythagoras' theorem, before moving on to proofs of many simple results often found in first-year undergraduate courses, such as common proofs of an introductory analysis course. Regardless of how simple or complex the proof may be, the author does not stray from his approach of showing a well-presented proof and then breaking it down statement by statement to explain and analyse the techniques that have been used. However, the proofs presented throughout the text are in no systematic order, which is why the author has produced a useful appendix full of proofs and explanations suitable for many first-year courses, such as linear algebra.

Given the immediate introduction to a formal mathematics layout, the text may require some persistence for a new undergraduate, but the author's approach should help, and the large number of exercises definitely will reinforce the techniques. Usefully, there are solutions to some exercises at the back of the book and more solutions are also available on the publisher's website. This text is largely successful in meeting its aims, and thus it is an ideal introduction for the first-year undergraduate mathematician.

Who is it for? First-year mathematics undergraduates.

Presentation: Formally presented with clear explanations and a useful appendix.

Would you recommend it? It provides an ideal grounding for any mathematics student.

Highly recommended

A Modern Introduction to Linear Algebra

Author: Henry Ricardo

Edition: First

Publisher: CRC Press/Taylor and Francis

Pages: 670

Price: £49.99

ISBN 9781439800409

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