Enhanced descriptions from Syndetics:

The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on 'Conjecture and Proof'. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of e, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader.

Table of contents provided by Syndetics

• Introduction (p. v)
• Part I. Proofs of Impossibility, Proofs of Nonexistence (p. 1)
• 1. Proofs of Irrationality (p. 3)
• 2. The Elements of the Theory of Geometric Constructions (p. 11)
• 3. Constructible Regular Polygons (p. 17)
• 4. Some Basic Facts About Linear Spaces and Fields (p. 21)
• 5. Algebraic and Transcendental Numbers (p. 27)
• 6. Cauchy's Functional Equation (p. 33)
• 7. Geometric Decompositions (p. 39)
• Part II. Constructions, Proofs of Existence (p. 47)
• 8. The Pigeonhole Principle (p. 49)
• 9. Liouville Numbers (p. 55)
• 10. Countable and Uncountable Sets (p. 59)
• 11. Isometries of R[superscript n] (p. 67)
• 12. The Problem of Invariant Measures (p. 75)
• 13. The Banach-Tarski Paradox (p. 81)
• 14. Open and Closed Sets in R. The Cantor Set (p. 85)
• 15. The Peano Curve (p. 93)
• 16. Borel Sets (p. 97)
• 17. The Diagonal Method (p. 103)
• References (p. 107)
• Hints (p. 109)
• Index (p. 117)

Reviews provided by Syndetics

CHOICE Review

Rigorous proof forms the backbone of higher mathematics. On the traditional model, students first encounter proof in high school geometry, then build on this exposure as they study calculus. These days, however, one cannot expect students to have had a proper high school geometry course, and many math departments have bled calculus courses of rigor for the sake of inclusivity. Accordingly, two decades ago, new "transition to abstraction" courses appeared to remediate the deficiencies caused by these changes. Sadly, most works that serve such courses have a limp, schoolmarmish quality; they try to present the apparatus of proof separated from any substantial mathematical content. Here, Laczkovich (Eotvos Lorand Univ., Hungary), famous for his amazing solution of Tarski's circle-squaring problem, introduces theorem proving using some of mathematics' towering achievements as his vehicle, e.g., the transcendence of e and the Banach-Tarski paradox. Thus the reader can learn not just the mechanics of proof but also its significance and its savor. Better still, this slender volume will initiate students into major themes of analysis, abstract algebra, topology, geometry, and number theory. General readers; lower- and upper-division undergraduates. D. V. Feldman University of New Hampshire