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.
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
The Universal Set
The Intersection of Sets
Union of Sets
Partitioning of Sets
Subtraction of Sets
The Algebra of Sets
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?
The Countable Set —The Smallest of the Infinite 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
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
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
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!
Exercises and Examples
- No. of pages:
- © Academic Press 1968
- 1st January 1968
- Academic Press
- eBook ISBN: