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The Art of Proof: Basic Training for Deeper Mathematics

Matthias Beck and Ross Geoghegan
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Tom Schulte
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This book offers an approach well-balanced between rigor and clarifying simplification. Dilbert and Foxtrot cartoons with philosophical quotes presage the introduction of axioms and preliminary propositions. This graceful and witty blend succeeds well in a textbook for a post-calculus course transitioning a student to higher mathematics. The Art of Proof can also well serve independent readers looking for a solitary path to a vista on higher mathematics. I wish that as a high school student with a single semester of calculus behind me I had discovered this book and taken it in and then taken it with me on my university journeys.

At well under two hundred pages, this text seems to be light and cursory. But do not let that belie the depth of content contained therein. There is thoroughness and breadth to this work. It is, basically, a tripartite work in Discrete, Continuous and Further Topics. Discrete builds from axioms of the integers on to natural numbers and then higher concepts of recursion, implication and the Well-Ordering Principle. Also covered are set theory basics, equivalence relations, and modular arithmetic. Continuous begins with axioms of the reals, building on that to bijection, injection and surjection as well as limits, completeness, rational and irrational numbers. Cardinality is used to explore the varieties of infinity. The Further Topics additional material is a combination of putting previous topics together as well as exploring new areas. Herein lie Cayley graphs, Lie groups, and generating functions. Also explored are complex numbers, continuity, and Euclidean geometry.

Axioms, theorems, and proofs are taught in a comfortable blend of discussion of the mathematics and motivating projects. This text could work for collegiate and advanced high school math clubs that I have participated in. The structure of the text supports self-direction. Each chapter, planned for a week’s study, ends with a direct reminder to consider the material slowly, each sentence at a time. Apply this advice to this well-written text and understanding will follow.

Tom Schulte teaches college algebra at Oakland Community College in Michigan and looks forward to enjoying more titles in the Springer series Undergraduate Texts in Mathematics.

Preface.- Notes for the Student.- Notes for Instructors.- Part I: The Discrete.- 1 Integers.- 2 Natural Numbers and Induction.- 3 Some Points of Logic.- 4 Recursion.- 5 Underlying Notions in Set Theory.- 6 Equivalence Relations and Modular Arithmetic.- 7 Arithmetic in Base Ten.- Part II: The Continuous.- 8 Real Numbers.- 9 Embedding Z in R.- 10. Limits and Other Consequences of Completeness.- 11 Rational and Irrational Numbers.- 12 Decimal Expansions.- 13 Cardinality.- 14 Final Remarks.- Further Topics.- A Continuity and Uniform Continuity.- B Public-Key Cryptography.- C Complex Numbers.- D Groups and Graphs.- E Generating Functions.- F Cardinal Number and Ordinal Number.- G Remarks on Euclidean Geometry.- List of Symbols.- Index.