The art of proof: basic training for deeper mathematics

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 teamsof students, either in class or in a mathematics club setting. These include:continuity, cryptography, groups, complex numbers, ordinal number, and generating functions. "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 coursesPrimarily 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 INDICE: Preface.- Notes for the Student.- Notes for Instructors.- Part I: The Discrete.- 1 Integers.- 2 Natural Numbers and Induction.- 3 Some Points ofLogic.- 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 Consequencesof 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.

• ISBN: 978-1-4419-7022-0
• Editorial: Springer
• Encuadernacion: Cartoné
• Páginas: 182
• Fecha Publicación: 29/08/2010
• Nº Volúmenes: 1
• Idioma: Inglés