Conjecture and Proof

Rating:
Summary: The basics of proof techniques covered in sufficient depth
Review: As the title indicates, the legendary Paul Erdos was involved in the creation of this book. In 1983, Erdos and other Hungarian mathematicians started the Budapest Semester in Mathematics (BSM), a program for American and Canadian undergraduate students. One of the courses in this program involves creative problem solving, which was the motivation for the material in this book. As is the case with books on problem solving, no particular area of mathematics is examined. The emphasis is on proof techniques, which are largely independent of the mathematical topic.
Of course, the quality of any book of this type is largely dependent on the choice of problems that are described, and in this case the chosen topics are excellent. The book is split into two main sections, which are further split into the following subsections:

I) Proofs of Impossibility, Proofs of Nonexistence.
1) Proofs of Irrationality.
2) The Elements of the Theory of Geometric Constructions.
3) Constructible Regular Polygons.
4) Some Basic Facts About Linear Spaces and Fields.
5) Algebraic and Transcendental Numbers.
6) Cauchy's Functional Equation.
7) Geometric Decompositions.

II) Constructions, Proofs of Existence.
8) The Pigeonhole Principle.
9) Liouville Numbers.
10) Countable and Uncountable Sets.
11) Isometries of R^n.
12) The Problem of Invariant Measures.
13) The Banach-Tarski Paradox.
14) Open and Closed Sets in R. The Cantor Set.
15) The Peano Curve.
16) Borel Sets.
17) The Diagonal Method.

While each of these topics is introduced, that does not mean that the coverage is superficial. The book is advertised as having more than elementary coverage, and I concur with that assessment. Detailed proofs of the main ideas are included with exercises at the end of each section. Hints for the solution of many of the problems are included in an appendix.
This is an excellent short introduction to many of the proof techniques that are the staple of working mathematicians. I strongly recommend it as a primary or secondary text for any course where the goal is to teach basic proof techniques to advanced undergraduates.