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**An accompaniment to higher mathematics.**
*(English)*
Zbl 0839.00004

Undergraduate Texts in Mathematics. New York, NY: Springer-Verlag. xvii, 198 p. (1996).

This book prepares undergraduate mathematics students to read mathematics independently and to understand and write proofs. It can serve as a supplement to any of the standard texts for a traditional “content” course in real analysis, abstract algebra, or topology.

The book begins by teaching how to read mathematics actively, constructing examples, extreme cases and non-examples to aid in understanding an unfamiliar theorem or definition (a technique familiar to any mathematician, but rarely taught); it provides practice by indicating explicitly where work with pencil and paper must interrupt reading. Then the book analyzes various proof forms (direct proof, proof by contraposition, proof by contradiction, proof by cases, proof by induction). Further the author turns to formal logical language and shows how to use it in mathematics (especially quantifier arguments). The common introductory material (such as sets, functions, relations and so on) is used for the numerous exercises, and the book concludes with a set of “Laboratories” on these topics in which the student can practice the skills learned in the earlier chapters.

The book begins by teaching how to read mathematics actively, constructing examples, extreme cases and non-examples to aid in understanding an unfamiliar theorem or definition (a technique familiar to any mathematician, but rarely taught); it provides practice by indicating explicitly where work with pencil and paper must interrupt reading. Then the book analyzes various proof forms (direct proof, proof by contraposition, proof by contradiction, proof by cases, proof by induction). Further the author turns to formal logical language and shows how to use it in mathematics (especially quantifier arguments). The common introductory material (such as sets, functions, relations and so on) is used for the numerous exercises, and the book concludes with a set of “Laboratories” on these topics in which the student can practice the skills learned in the earlier chapters.

Reviewer: I.N.Lushchakova (Minsk)