Format:
Online-Ressource (XXII, 182 p. 23 illus, digital)
ISBN:
9781441970237
,
1280391383
,
9781280391385
Series Statement:
Undergraduate Texts in Mathematics 0
Content:
The Discrete -- Integers -- Natural Numbers and Induction -- Some Points of Logic -- Recursion -- Underlying Notions in Set Theory -- Equivalence Relations and Modular Arithmetic -- Arithmetic in Base Ten -- The Continuous -- Real Numbers -- Embedding Z in R -- Limits and Other Consequences of Completeness -- Rational and Irrational Numbers -- Decimal Expansions -- Cardinality -- Final Remarks -- Further Topics -- Continuity and Uniform Continuity -- Public-Key Cryptography -- Complex Numbers -- Groups and Graphs -- Generating Functions -- Cardinal Number and Ordinal Number -- Remarks on Euclidean Geometry.
Content:
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.
Note:
Includes index
,
""The Art of Proof""; ""Copyright""; ""Preface""; ""Contents""; ""Notes for the Student""; ""Notes for Instructors""; ""Part I: The Discrete""; ""Chapter 1 Integers""; ""1.1 Axioms""; ""1.2 First Consequences""; ""1.3 Subtraction""; ""1.4 Philosophical Questions""; ""Chapter 2 Natural Numbers and Induction""; ""2.1 Natural Numbers""; ""2.2 Ordering the Integers""; ""2.3 Induction""; ""2.4 The Well-Ordering Principle""; ""Chapter 3 Some Points of Logic""; ""3.1 Quantifiers""; ""3.2 Implications""; ""3.3 Negations""; ""3.4 Philosophical Questions""; ""Chapter 4 Recursion""; ""4.1 Examples""
,
""4.2 Finite Series""""4.3 Fishing in a Finite Pool""; ""4.4 The Binomial Theorem""; ""4.5 A Second Form of Induction""; ""4.6 More Recursions""; ""Chapter 5 Underlying Notions in Set Theory""; ""5.1 Subsets and Set Equality""; ""5.2 Intersections and Unions""; ""5.3 Cartesian Products""; ""5.4 Functions""; ""Chapter 6 Equivalence Relations and Modular Arithmetic""; ""6.1 Equivalence Relations""; ""6.2 The Division Algorithm""; ""6.3 The Integers Modulo n""; ""6.4 Prime Numbers""; ""Chapter 7 Arithmetic in Base Ten""; ""7.1 Base-Ten Representation of Integers""
,
""7.2 The Addition Algorithm for Two Nonnegative Numbers (Base 10)""""Part II: The Continuous""; ""Chapter 8 Real Numbers""; ""8.1 Axioms""; ""8.2 Positive Real Numbers and Ordering""; ""8.3 Similarities and Differences""; ""8.4 Upper Bounds""; ""Chapter 9 Embedding Z in R""; ""9.1 Injections and Surjections""; ""9.2 The Relationship between Z and R""; ""9.3 Apples and Oranges Are All Just Fruit""; ""Chapter 10 Limits and Other Consequences of Completeness""; ""10.1 The Integers Are Unbounded""; ""10.2 Absolute Value""; ""10.3 Distance""; ""10.4 Limits""; ""10.5 Square Roots""
,
""Chapter 11 Rational and Irrational Numbers""""11.1 Rational Numbers""; ""11.2 Irrational Numbers""; ""11.3 Quadratic Equations""; ""Chapter 12 Decimal Expansions""; ""12.1 Infinite Series""; ""12.2 Decimals""; ""Chapter 13 Cardinality""; ""13.1 Injections, Surjections, and Bijections Revisited""; ""13.2 Some Countable Sets""; ""13.3 Some Uncountable Sets""; ""13.4 An Infinite Hierarchy of Infinities""; ""13.5 Nondescribable Numbers""; ""Chapter 14 Final Remarks""; ""Further Topics""; ""Appendix A Continuity and Uniform Continuity""; ""A.1 Continuity at a Point""
,
""A.2 Continuity on a Subset of R""""A.3 Uniform Continuity""; ""Appendix B Public-Key Cryptography""; ""B.1 Repeated Squaring""; ""B.2 Diffieâ€?Hellman Key Exchange""; ""Appendix C Complex Numbers""; ""C.1 Definition and Algebraic Properties""; ""C.2 Geometric Properties""; ""Appendix D Groups and Graphs""; ""D.1 Groups""; ""D.2 Subgroups""; ""D.3 Symmetries""; ""D.4 Finitely Generated Groups""; ""D.5 Graphs""; ""D.6 Cayley Graphs""; ""D.7 G as a Group of Symmetries of Î?""; ""D.8 Lie Groups""; ""Appendix E Generating Functions""; ""E.1 Addition""; ""E.2 Multiplication and Reciprocals""
,
""E.3 Differentiation""
Additional Edition:
ISBN 9781441970220
Additional Edition:
Buchausg. u.d.T. Beck, Matthias, 1970 - The art of proof New York : Springer, 2010 ISBN 9781441970220
Language:
English
Subjects:
Mathematics
Keywords:
Mathematische Methode
;
Mathematische Methode
;
Mathematik
;
Beweis
;
Lehrbuch
;
Einführung
DOI:
10.1007/978-1-4419-7023-7
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
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