Stories About Sets

1st Edition - January 1, 1968
Author: N. Ya. Vilenkin
Editors: D. Allan Bromley, Nicholas Declaris, W. Magnus
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 1 4 3 5 - 1

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.… Read more

Purchase options

LIMITED OFFER

Save 50% on book bundles

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

Institutional subscription on ScienceDirect

Request a sales quote

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.

ForewordPreface1. Some Extraordinary Properties of Infinite Sets The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet From the Author2. Sets and Operations on Sets What Do We Mean by a Set? How We Specify a Set To Shave or Not to Shave? The Empty Set The Theory of Sets and Elementary Mathematics Subsets The Universal Set The Intersection of Sets Union of Sets Partitioning of Sets Subtraction of Sets The Algebra of Sets Boolean Algebras3. The Cardinality of Sets Equality Between Sets On the Dance Floor For Every Flow There Is an Ebb Can a Part Be Equal to the Whole? Countable Sets Algebraic Numbers Unequal Sets The Countable Set —The Smallest of the Infinite Sets Uncountable Sets The Census That Never Took Place The Uncountability of the Continuum The Existence of Transcendental Numbers Long and Short Line Segments Have Equally Many Points Segment and Square Somehow One Problem Does Not Work Out Is There a Set of Largest Cardinality? The Arithmetic of the Infinite Infinite Exponents On the Ordering of Numbers Completely Ordered Sets The Enigmatic Axiom Two Apples from One4. Remarkable Functions and Curves, or a Stroll through a Mathematical Art Museum How the Notion of Function Developed The Genie Escapes from the Bottle Wet Points The Devil's Staircase A Prickly Curve A Closed Curve of Infinite Length A Mathematical Carpet Euclid Does Not Rely on Euclid Are Rigorous Definitions Needed? A Curve Is the Path of a Moving Point The Theorem Is Obvious, but the Proof Is Not A Curve Passing through All the Points of a Square Everything Had Come Unstrung How to Make a Statue Continua Cantor Curves Can the Area of a Curve Be Different from Zero? Domains without Area Some Surprising Examples Domains and Boundaries The Great Irrigation Project A "Nondissertable" Subject The Inductive Definition of Dimension The Article Is to Be Printed, Not Reviewed! Conclusion Exercises and Examples