# Stories About Sets

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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