Stories About Sets

Stories About Sets

1st Edition - January 1, 1968

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  • Author: N. Ya. Vilenkin
  • eBook ISBN: 9781483214351

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Description

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.

Table of Contents


  • Foreword

    Preface

    1. Some Extraordinary Properties of Infinite Sets

    The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet

    From the Author

    2. 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 Algebras

    3. 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 One

    4. 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

Product details

  • No. of pages: 166
  • Language: English
  • Copyright: © Academic Press 1968
  • Published: January 1, 1968
  • Imprint: Academic Press
  • eBook ISBN: 9781483214351

About the Author

N. Ya. Vilenkin

About the Editors

D. Allan Bromley

Nicholas Declaris

W. Magnus

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