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Mathematics | The Art of Proof

The Art of Proof

Basic Training for Deeper Mathematics

Beck, Matthias, Geoghegan, Ross

2010, XXII, 182 p. 23 illus.

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  • Presents fundamental mathematics, integers and real numbers, in a way that asks for student participation, while teaching how mathematics is done
  • Provides students with methods and ideas they can use in future courses
  • Primarily for: undergraduates who have studied calculus or linear algebra; mathematics teachers and teachers-in-training; scientists and social scientists who want to strengthen their command of mathematical methods
  • Extra topics in appendices give instructor flexibility
The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. Some of the proofs are presented in detail, while others (some with hints) may be assigned to the student or presented by the instructor. The authors recommend that the two parts of the book -- Discrete and Continuous -- be given equal attention. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Content Level » Lower undergraduate

Keywords » Cardinal number - Countable set - algebra - cardinality - completeness of R - high-order recursions - integers modulo n - noncomputable numbers - set theory - strong induction - universal quantifiers - well ordering principle

Related subjects » Algebra - Analysis - Mathematics

Table of contents 

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.

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