Vilenkin / Bromley / Declaris Stories About Sets

1. Auflage 2014
ISBN: 978-1-4832-1435-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 166 Seiten, Web PDF

Stories About Sets discusses the cardinality of sets and mathematical concepts, such as function, curve, surface, dimensions, and the paradoxical properties of curves and surfaces. The book reviews sets, operations on sets, the empty set, subsets, the universal sets, intersection of sets, union of sets, partitioning of sets, and boolean algebras. The text also discusses the cardinality of sets, including equality between sets, countable sets, unequal sets, the uncountability of the continuum, the existence of transcendental numbers, and the enigmatic axiom. The book analyzes if a part can be equal to the whole (which turns out to be true if it is applied to infinite sets). The text also discusses the arithmetic of the infinite such as involving the multiplication of infinite cardinalities. The book explains some remarkable functions and curves, the Dirichlet's function, Cantor's set, points of fracture, and continuous functions whose graphs possess a tangent at no point. The text shows how to construct a closed curve of infinite length or a curve passing through all the points of a square. The book can prove interesting and highly educational for students with mathematic or algebra subjects, as well as for academicians involved in teaching statistics or mathematics.

Autoren/Hrsg.

Vilenkin, N. Ya.

Bromley, D. Allan

Declaris, Nicholas

Magnus, W.

Inhaltsverzeichnis

1;Front Cover;1
2;Stories About Sets;4
3;Copyright Page;5
4;Table of Contents;12
5;Foreword;6
6;Preface;8
7;Chapter 1. Some Extraordinary Properties of Infinite Sets;16
7.1;The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet;19
7.2;From the Author;29
8;Chapter 2. Sets and Operations on Sets;31
8.1;What Do We Mean by a Set ?;31
8.2;How We Specify a Set;33
8.3;To Shave or Not to Shave ?;36
8.4;The Empty Set;39
8.5;The Theory of Sets and Elementary Mathematics;41
8.6;Subsets;42
8.7;The Universal Set;44
8.8;The Intersection of Sets;44
8.9;Union of Sets;46
8.10;Partitioning of Sets;50
8.11;Subtraction of Sets;52
8.12;The Algebra of Sets;54
8.13;Boolean Algebras;56
9;Chapter 3. The Cardinality of Sets;58
9.1;Equality between Sets;58
9.2;On the Dance Floor;59
9.3;For Every Flow There Is an Ebb;61
9.4;Can a Part Be Equal to the Whole ?;62
9.5;Countable Sets;64
9.6;Algebraic Numbers;66
9.7;Unequal Sets;68
9.8;The Countable Set—The Smallest of the Infinite Sets;71
9.9;Uncountable Sets;72
9.10;The Census That Never Took Place;73
9.11;The Uncountability of the Continuum;76
9.12;The Existence of Transcendental Numbers;78
9.13;Long and Short Line Segments Have Equally Many Points;79
9.14;Segment and Square;81
9.15;Somehow One Problem Does Not Work Out;84
9.16;Is There a Set of Largest Cardinality?;86
9.17;The Arithmetic of the Infinite;88
9.18;Infinite Exponents;91
9.19;On the Ordering of Numbers;93
9.20;Completely Ordered Sets;95
9.21;The Enigmatic Axiom;97
9.22;Two Apples from One;99
10;Chapter 4. Remarkable Functions and Curves, or a Stroll through a Mathematical Art Museum;101
10.1;How the Notion of Function Developed;101
10.2;The Genie Escapes from the Bottle;106
10.3;Wet Points;108
10.4;The Devil's Staircase;112
10.5;A Prickly Curve;114
10.6;A Closed Curve of Infinite Length;119
10.7;A Mathematical Carpet;122
10.8;Euclid Does Not Rely on Euclid;126
10.9;Are Rigorous Definitions Needed?;127
10.10;A Curve Is the Path of a Moving Point;129
10.11;The Theorem Is Obvious, but the Proof Is Not;133
10.12;A Curve Passing through All the Points of a Square;135
10.13;Everything Had Come Unstrung;137
10.14;How to Make a Statue;139
10.15;Continua;141
10.16;Cantor Curves;143
10.17;Can the Area of a Curve Be Different from Zero ?;144
10.18;Domains without Area;148
10.19;Some Surprising Examples;150
10.20;Domains and Boundaries;152
10.21;The Great Irrigation Project;154
10.22;A "Nondissertable" Subject;156
10.23;The Inductive Definition of Dimension;159
10.24;The Article Is to Be Printed, Not Reviewed!;161
10.25;Conclusion;164
10.26;Exercises and Examples;165