The Elements of Cantor Sets: With Applications

THE ELEMENTS OF CANTOR SETS— WITH APPLICATIONS

THE ELEMENTS OF CANTOR SETS— WITH APPLICATIONS ROBERT W. VALLIN Slippery Rock University

WILEY

Copyright © 2013 by John Wiley & Sons, Inc. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Vallin, Robert W. The elements of Cantor sets : with applications / Robert W. Vallin, Department of Mathematics, Slippery Rock University, Slippery Rock, PA. — First edition, pages cm Includes bibliographical references and index. ISBN 978-1-118-40571-0 (hardback) 1. Cantor sets. 2. Measure theory. 3. Mathematical analysis. 4. Cantor, George, 1941- I. Title. QA612.V35 2013 515'.8—dc23 2013009452 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1

To Micah, Jessica, Rachel, Sophie, and Sean. And to Jackie, who always believes in me.

CONTENTS IN BRIEF

1 A Quick Biography of Cantor

1

2 Basics

5

3 Introducing the Cantor Set

17

4 Cantor Sets and Continued Fractions

51

5 p-adic Numbers and Valuations

67

6 Self-Similar Objects

91

7 Various Notions of Dimension

119

8 Porosity and Thickness-Looking at the Gaps

143

9 Creating Pathological Functions via C

157

10 Generalizations and Applications

181

11 Epilogue

217

VII

CONTENTS

Foreword

xiii

Preface

xv

Acknowledgments

xvii

Introduction

xix

1

A Quick Biography of Cantor

1

2

Basics

5

2.1

3

Review Exercises

5 14

Introducing the Cantor Set

17

3.1 3.2

17 21 22 29 36

Some Definitions and Basics Size of a Cantor Set 3.2.1 Cardinality 3.2.2 Category 3.2.3 Measure

ix

X

CONTENTS

3.3

4

5

51

4.1 4.2 4.3 4.4

52 59 60 63 65

8

Introducing Continued Fractions Constructing a Cantor Set Diophantine Equations Miscellaneous Exercises

p-adic Numbers and Valuations

67

5.1 5.2

67 72 72 75 80 82 88

Some Abstract Algebra p-adic Numbers 5.2.1 An Analysis Point of View 5.2.2 An Algebra Point of View p-adic Integers and Cantor Sets p-adic Rational Numbers Exercises

Self-Similar Objects 6.1 6.2 6.3 6.4 6.5

7

47 48

Cantor Sets and Continued Fractions

5.3 5.4

6

Large and Small Exercises

The Meaning of Self-Similar Metric Spaces Sequences in (S, d) Affine Transformations An Application for an IFS Exercises

91 91 92 97 106 114 116

Various Notions of Dimension

119

7.1 7.2 7.3 7.4 7.5 7.6

119 123 127 128 131 136 140

Limit Supremum and Limit Infimum Topological Dimension Similarity Dimension Box-Counting Dimension Hausdorff Measure and Dimension Miscellaneous Notions of Dimension Exercises

Porosity and Thickness-Looking at the Gaps

143

8.1

143

The Porosity of a Set

CONTENTS

8.2 8.3 8.4 8.5 8.6 8.7

9

10

11

146 149 150 151 153 154 156

Creating Pathological Functions via C

157

9.1 9.2 9.3 9.4 9.5 9.6

157 161 168 171 173 177 180

Sequences of Functions The Cantor Function Space-Filling Curves Baire Class One Functions Darboux Functions Linearly Continuous Functions Exercises

Generalizations and Applications

181

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13

181 185 186 193 195 197 198 201 203 205 207 209 211

Generalizing Cantor Sets Fat Cantor Sets Sums of Cantor Sets Differences of Cantor Sets Products of Cantor Sets Cantor Target Ana Sets Average Distance Non-Averaging Sets Cantor Series and Cantor Sets Liouville Numbers and Irrationality Exponents Sets of Sums of Convergent Alternating Series The Monty Hall Problem

Epilogue

References Index

Symmetric Sets and Symmetric Porosity A New and Different Definition of Cantor Set Thickness of a Cantor Set Applying Thickness A Bit More on Thickness Porosity in a Metric Space Exercises

XI

217 219

FOREWORD

Among the delights of mathematics are those moments when it surprises, when something unexpected emerges from a seemingly mundane construction that is pushed beyond the bounds of the finite: staircases of finite length for which one can never climb from one step to the next, continuous and bounded functions that can be in­ tegrated but whose integral does not have a derivative, space-filling curves and frac­ tional dimensions, and the Banach-Tarski paradox: the ability - in theory - to cut a pea into finitely many pieces and reassemble those using only rigid motions (ro­ tations and translations) into a solid object the size of the sun. Such mathematical surprises are the stuff of which great mathematical advances are made. They chal­ lenge us to broaden our understanding of the world in which we live. The Cantor set provides one of these surprising moments. If we take an interval of the real number line, say from 0 to 1, and remove finitely many little intervals from it, we are left with finitely many little intervals. No surprise here. But what if we remove infinitely many little intervals? Are we left with infinitely many little intervals? No less a mathematician than Axel Harnack (1851-88) implicitly assumed this obvious conclusion. He was very wrong. To see how wrong he was, imagine that from our interval [0,1] we remove ev­ ery real number with 7 in its decimal expansion. This takes out everything from 0.7 to just below 0.8, one interval. It also takes out the nine intervals [0.07, 0.08), [0.17,0.18), ..., [0.97,0.98). We are not done. We also have to take out the all of XIII

XIV

FOREWORD

the intervals of length 0.001 that start at a number with two digits that are not 7, followed by a 7, such as 0.237. That adds 92 intervals of length 1/103. This goes on forever. The total amount that we remove is 1

9

92

93

_ 1_ (

_9_

Io + io2 + To3 + To4 + ''' _ To V + i o

+

92

\ _ 1 /

1

\

T o 2 + ' " y ™ i o l^i-9/ioJ ~

It appears that we have removed the entire interval. But, in fact, it is quite easy to come up with numbers that are left behind. We still have 1/2 = 0.5. No 7s in that decimal expansion. We still have 1/3 = 0.333 In some sense, we still have as many real numbers as we started with. Who needs the symbol 7? If we represent our real numbers in base 9, the remaining nine symbols are sufficient to express every real number in [0,1]. Lots of numbers are left, but no intervals. You cannot get from 1/2 to 1/3 without crossing a number with a 7 in its decimal expansion. More than this, between any two distinct real numbers there will always be a real number with a 7 in its decimal expansion. What is left is a very unintuitive set that is so full of holes that nothing is left, and yet it still describes everything. It is what we call a Cantor Set. The name Cantor Set is a bit of a misattribution. Henry J.S. Smith (1826-83) used these sets in a paper published in 1875. Unfortunately, Smith was an Englishman at a time when few expected any significant mathematics to come out of England. Any really important mathematics would have been published in German. The next paper to use these sets was by Vito Volterra in 1881 (1860-1940). Volterra would come to be known as one of greatest of all mathematicians, but in 1881 he was a mere student publishing his results in an obscure Italian journal. No one noticed. When Georg Cantor rediscovered and described these sets in 1883, it was one piece of a major and highly visible assault on our understanding of the structure of the real number line. Now mathematicians paid attention. In this book, Robert Vallin leads us through the many surprises of Cantor Sets. This is only the beginning. The structure of these sets is echoed in much of 20th century mathematics from p-adic numbers to fractals. This tour will explore the connections to continued fractions, to self-similarity and non-integer dimensions, to derivatives that are not continuous and space-filling curves that are, and to one of my favorite mathematical curiosities, the Cantor function, more aptly known as The Devils Staircase. David M. Bressoud

PREFACE

Like probably all mathematicians, I cannot recall the first time I heard about the Cantor Set. I am sure I was an undergraduate, but I just don't remember. What I do know is that I have been fascinated by it since first coming across it. Start with the unit interval, remove what seems like everything, and the Cantor Set is what is left over. There are numerous points in it, despite the fact that so much has been removed. With disbelief I read that it has as many points as the interval [0,1]. Someone told me the number 1/4 is a point in the set. I probably just nodded up and down. I was sure I was being told the truth, but had no clue why it was true. In graduate school, I found out even more. It seemed like everywhere I turned there was the Cantor Set. In Analysis, it was an example of a measure zero set. In Topology, it was nowhere dense. In a class on fractals, it had non-integral dimension. It lived in the line, but had dimension that was neither 0 (like a point) nor 1 (like a line segment). It seemed to be everywhere I turned: analysis, topology, abstract algebra, probability, and the list goes on. It was cool, but I was busy getting my PhD. For awhile the Cantor Set went by the wayside. Once out of school, I gained the freedom to study whatever was my pleasure. Most of my years I've been an analyst/topologist. However, when articles involving the Cantor Set came along, I was sure to read them and enjoy myself. As a great believer in students graduating with a large collection of examples/counter-examples in their quivers, the Cantor Set was one of my go-to topics. xv

XVI

PREFACE

In 2011, I was approached by a student in my analysis course about an indepen­ dent study class the next semester. He was interested in finding out more about the Cantor Set. I was very excited about the idea. Once we got approval for the plan, I set forth to make an outline. The course was written pseudo-IBL (Inquiry-Based Learning). Every week we would meet and I would give him some definitions and some open questions. Sometimes these were theorems which required proofs; some­ times they were false statements that needed counterexamples. We covered a little bit of everything I could think of and the course went well. In putting my outline together I was surprised at how scattered my resources were. There was no central place that had all that I needed. Existing books were too nar­ rowly focused and/or too technical as were the websites around. It occurred to me that the outline was the beginning of something good. So I decided that I would be the person to put this book together. In writing this book, I walked a fine line: The text is not written very formally; but being a serious text, does have the form theorem/proof. However, in writing this manuscript for advanced undergraduates I left out many proof because they were too long and technical for this intended audience. It is my hope that anyone interested in the details will be able to use the bibliography to go to the source and find out the nitty-gritty. ROBERT W. VALLIN Slippery Rock, PA January 2013

ACKNOWLEDGMENTS

Any endeavor like this is not a work done alone, even though it is only my name on the cover. As noted in the Preface, without my student Andy Brown and his interest in an independent study on Cantor Sets, I would not have thought of putting together my notes which then became this manuscript. Those who lent an ear and encouraged me include Lyn Miller, Carol Schumacher, Rich Marchand, and Ron Taylor. Gary Grabner was a huge help with the figures. Rita McClelland and Kelsi Wren were able to find me whatever I needed on interlibrary loan. At Wiley, thanks go out to Susanne Steitz-Filler, my editor, Christine Punzo, for making this readable, and Amy Hendrickson, who patiently answered my ETijXquestions. My biggest cheerleader and fan was my wife, Jacqueline Jensen-Vallin. That said, any and all mistakes are mine and mine alone. If you find any errors, or if I have missed your favorite topic, please don't hesitate to contact me. R.W.V.

XVII

INTRODUCTION

This is not just about the Cantor Set. If it were, there would be much less to say. As an object itself, the set can be fascinating, but what is most appealing is not the set per se, but where the set can lead one. In analysis one uses the notions of measure, category, and cardinality to describe many objects. When teaching those topics the Cantor Set always pops up as an example to fortify understanding of the concept. The text makes for a good ancillary in several different courses: analysis (using Chapters 2, 3, 9 as a start), topology (Chapters 2, 8, 9), fractals (Chapters 6, 7, 10), and algebra (Chapters 2, 5, 4). Many schools are introducing the Capstone Experience Course to their curricu­ lum. This is a Senior-Level class that, while having a unifying theme, draws in material from the previous coursework for their undergraduates. These courses can include analysis, algebra, topology, and number theory. Several Capstone Courses also require an out-of-class project. This book has a unifying theme, of course, and shows how the Cantor Set appears in so many different mathematical topics, thus bringing together the collection of courses that make up the lion's share of the upperlevel math curriculum. Given the wide range of topics covered, there are plenty of places for students to find topics and questions to study outside the classroom for the project portion of the course. This manuscript, which arose from an independent study course, is an example of the type of book needed for an independent study. Cantor Sets are studied from many XIX

XX

INTRODUCTION

different perspectives, with multiple definitions and applications. The book is meant to be something a student can read on his/her own, meeting with a professor from time to time to discuss the topics. There are so many ideas covered that whether a student has a preference for analysis or algebra or something else, that subject can be found. If students enjoy applying what he/she learns, then Chapters 9 and 10 cover so many ideas that one can find a direction to take their projects. Lastly, this book is a resource for professors. As we all know, the "sage on the stage" reading from well-worn notes or overhead slides is an outdated idea. The pro­ fessor, being an expert in their field, should be able to supplement any text, be it with anecdotes, apocrypha, extra examples, generalizations, or applications. Parsed into its separate chapters and sections, this book can be used to add to several different courses in the undergraduate mathematics curriculum. It does have a purpose out­ side the classroom. With the topics covered and detailed bibliography, one can find a myriad of directions to take one's research.

CHAPTER 1

A QUICK BIOGRAPHY OF CANTOR

Georg Cantor Library of Congress, courtesy AIP Emilio Segr Visual Archives The Elements of Cantor Sets -With Applications, First Edition. By Robert W. Vallin Copyright © 2013 John Wiley & Sons, Inc.

1

2

A QUICK BIOGRAPHY OF CANTOR

Georg Ferdinand Ludwig Philipp Cantor was one of the most important mathe­ maticians of the last half of the 19th century. Born 3 March 1845 in St. Petersburg, Russia, he was the oldest of six children. Due to his father's ill health, the family moved to Frankfurt, Germany, hoping to find milder winters. In 1862 he started his higher schooling at the Federal Polytechnic Institute in Zurich. After the death of his father, Cantor moved on to study at the University of Berlin. Among his professors there were Kronecker, Weierstrass, and Kummer. In 1867, he earned his PhD for his thesis entitled "De aequationibus secundi gradus indeterminatis (On Indeterminate Equations of the Second Degree)." Cantor began teaching at the University of Halle, where he would spend his entire career, in 1869. He was promoted to full professor in 1879, which was an achieve­ ment at such a young age. However, Cantor wanted to move to the more presti­ gious University of Berlin, hoping to become chair of the department. This could not happen as Kronecker, who led the department at Berlin, was not supportive of Cantor and his work. Kronecker was one of the founders of the constructive view­ point; that means only dealing with mathematical objects that could be explicitly made. As an example of Kronecker's feelings on this, when von Lindemann proved the existence of transcendental numbers (a type of number defined later on in this book), Kronecker praised the proof, but also said the argument [19] "proved noth­ ing because there were no transcendental numbers." Much of Cantor's work dealt with sets whose members could be described and were well-defined, but not explicit. Kronecker went so far as to try and pressure Heine to not publish Cantor's paper "Uber trigonometrische Reihen" in the journal Mathematische Annalen. In a letter to Hermite, Cantor complained that Kronecker, among, other things, called his work "humbug." The pressure from the non-acceptance of his work would drive Cantor into periods of depression, which he suffered from until the end of his life. He carried on correspondence with some of the greater mathematicians of the day (e.g., Mittag-Leffler, Dedekind), but would at times cease communication. He was very sensitive to even well-intentioned criticisms of his work. In 1894, Cantor suffered his first attack of depression. He even turned his attention away from math­ ematics for a time, lecturing on philosophy and working on proving that Shakespeare did not write the plays attributed to him. Instead, Cantor believed they were written by Francis Bacon. Cantor was hospitalized again for depression in 1899 and then several more times starting in 1904. These sapped him of much of his zeal for mathematics. However, his work continued to be appreciated. In 1912 he was awarded an honorary doctorate from the University of St. Andrews in Scotland. He officially retired in 1913 and died on 6 January 1918 in a sanatorium where he had been for the last year of his life. Cantor's contributions to mathematics are vast, with results in number theory (his thesis topic) and set theory (which he founded). Some of his most well-known ideas are the Cantor Set, the notion of different types of infinities, and the Continuum Hy­ pothesis. Working on the infinite drew the ire of some mathematicians, philosophers, and religious scholars. Philosophers felt there were too many contradictions in deal­ ing with the infinite. For example, if a and b are two positive numbers, then a < a + b and b < a + b. However, oo < a + oo is not true. No less a mathematician than

3

Poincare said that in the future people would look on Cantor's work as "a disease from which one has recovered." To quote from [19]: Christian theologians were also opposed to the actual infinite; for the most part they regarded the idea as a direct challenge to the unique and absolutely infinite nature of God.

However, Cantor was not without his supporters. At the Second International Congress held at the Paris World Exposition of 1900, David Hilbert presented a list of the major unsolved problems of the time, hoping to spur interest in what he believed to be the most important problems of the day. The item at the top of the list was Cantor's Continuum Hypothesis. Hilbert is famously quoted as saying, "No one will drive us from the paradise which Cantor created for us." Hilbert's influence was profound, as can be seen in the vast list of papers written just in the last few years that mention Cantor, Cantor Sets, Cantor Measure, Cantor Spaces, and more.

CHAPTER 2

BASICS

This is the "review" chapter. That means much of what we look at here the reader is familiar with. Thus there will not be a lot of proofs given. Most of these concepts are familiar, even if a bit of dust must be shaken off. Hopefully those ideas which are unfamiliar will be few and easily digested by the reader.

2.1

Review

This chapter begins with sets. A set is a well-defined collection of objects. We will use capital letters to denote a set. There are two typical ways to write a set: List Notation and Set-Builder Notation. As the name suggests, List Notation means the writer lists (some of) the objects in the set. Note that sets are written using braces for enclosure. For example, A = {a, b, c, d} or B = { 1 , 2 , 3 , . . . , 9,10}, where the ellipses (...). means "continue in this manner." For Set-Builder Notation, rather than listing the items in the set, the members are described by some proposition, P(x). This is a useful way of doing things as some sets defy listing. An example of this notation is C — {x : x is a letter in the English Alphabet} The Elements of Cantor Sets -With Applications, First Edition. By Robert W. Vallin Copyright © 2013 John Wiley & Sons, Inc.

5

6

BASICS

which is of the form {x : P(x)}. The colon is usually translated as "such that" or "so that," and then followed by a description. This is read as, "C is the set of x such that x is a letter in the English Alphabet." The empty set, 0, is the set with no elements in it. The textbook definition, which looks a bit strange, is 0 = {a : a ^ a}. An object which is a member of a set is called an element of the set and usually written with a lowercase letter. The symbol for is an "element" of is s. Rather than write, "a is a member of the set X," we just write "a G X." If an object is not an element of the set, then we use the symbol ^ as in t x £ B, where => is read as "implies." Using an empty argument, it is possible to prove that for every set B,$ C B. Subsets are used to prove equality of two sets. The sets A and B are equal if and only if A C B and B C A. If we know that A C B and A y^ B, then we say A is a proper subset of B which we write as A C B. Given two sets A and B, there are several ways to create new sets. The union of A and B is the set defined by AU B = {x :x € Aorx

€ B}.

The intersection of ^4 and 5 is the set defined by AnB

= {x : x £ A and x e B}.

If two sets have no element in common (A P\ B = 0), then we say A and B are disjoint. A collection of sets1 {An} is called pairwise disjoint if for i 7^ j we have A, n A,- = 0. The se/ difference of .A and Z? is the set defined by A \ B = {x : x € A and x g B}. A specific set difference which we will encounter many times is the complement of A. This is

Ac = U\A,

where U represents the Universal Set, the collection of all object under consideration. For much of this text U = K. Relating all these operations are De Morgan's Laws: (AUB)C

=ACDBC,

1 We could refer to this as a set of sets; however, there can be issues with this if we are not precise. To learn more on set theory, we suggest [21 ] or [36].

REVIEW

and

(AC\B)C

7

=Acl)Bc.

Union, intersection, and DeMorgan's Laws can be generalized to countably many sets A\,Az, A3, In this situation, UjAi = {x : there exists an i such that x € Ai} and HiAi = {x : for all i, x € At}, while DeMorgan's Laws become ( U ^ ) C = ni(Af)

and (HiAif

= Ut(A?)■

Finally, the product of non-empty ^4 and B is the set of all ordered pairs of elements from A and B, respectively; that is, Ax B = {(a,b)

:a€A,beB}.

If A or B is empty, then A x B = 0. fl

EXAMPLE 2.1 Let A = {a, 6, c} and I? = {a; : x is an English vowel} = {a, e, i, o, u} where the Universal Set is the English Alphabet, U = {a, b, c, d,..., y, z}. Then ■ A U B = {a, b, c, e, i, o, u}. • AC\B = {a}. ■ A\B

= {b,c}.

• Ac = {d,e: / , . . . , y,z}. • Bc = {x : x is an English consonant}. ■ AxB = {(a, a), (a, e), (a, i), (a, o), (a, it), (b, a), (b, e), (6, i), (b, o), (b, it), (c, a). (c,e),(c,i),(c,o),(c,u)}.

■

Specific sets to know include ■ The Natural Numbers: N = { 1 , 2 , 3 , . . . , }. ■ The Integers: Z = {..., - 2 , - 1 , 0 , 1 , 2 , . . . } . ■ The Rational Numbers: Q — {x : x — p/q, where p,q &Z,q j^O}. • The Real Numbers: R = (—00,00).

8

BASICS

Let a . i e K . We shall look at some specific sets in the real line. Intuitively, a set A in R is bounded if there are numbers m and M where no element of A is below m or above M. Now we define intervals as ■ Closed and bounded: [a, b] = {x G R : a < x < b}. ■ Open and bounded: (a, b) = {x G R : a < x < b}. • Open and unbounded: (a, oo) = {x G R : a < x} or (—00, b) = {x G R : x < b}. • Closed and unbounded: [a, 00) = {x G R : a < x} or (—00, 6] = {x G R : x < b}. Generalizing open and closed intervals leads us to open and closed sets in the line. A set A is open if for every x G A, there is an open interval /, such that x G / C A. A set B is closed its complement R \ B is open. Now that we have sufficiently clear ideas of open and closed sets, we can further our understanding of both the interior and boundary of a set. Formally, a: is a point in the boundary of A (written as dA) if every open interval containing x also contains points in both A and Ac. The interior of the set (written A°) can be expressed as A \ dA. Another way to define this is x G A° if there exists an open interval I such that x € / C A. Lastly, for any set A, the closure of A, A is the set along with its boundary, A = A U dA. S

EXAMPLE 2.2 Let A = (0,1) U (2, 00), B = [0,1] and C = Q n (0,1). Then A is open (but not an open interval), B is closed, C is neither open nor closed. dA = {0,1, 2}, d_B = {0,1}, dC = JO, 1]. Meanwhile, A0 = A, B° = (0,1), and C° = 0 and A=[0,l]U[2,oo),B = B = C. u

Starting with the open sets, we will create what is called a a-algebra. A non­ empty collection of sets X is called a and any open set is Qs- The interval (0,1) is an example of an J v set that is not a closed set since (0,l) = U neN [0 + l / n , l - l / n ] . Our first theorem, relates Ta and Qs sets and their complements. Theorem 2.1 If A is a Qs set in R, then Ac is an T„ set in R.

REVIEW

9

Proof: This is just an application of the generalized DeMorgan's Law. If A is a Gs set, we can write A = n(Oj) where each Oi is an open set. Then

AC = (HOif = U;(Of). Now each Of is a closed set, thus Ac is Fa. 4k Let A and 5 be sets. A function f : A —> B is a rule which assigns to each element in A exactly one element in B. This means / = {( a , b) : a € A, b € B, and (a, 6) € /, (a, 6') G / implies b = b'}. Of course, usually we do not write (a, b) s / , but instead / ( a ) = b. The set .A is called the domain of / and B is the co-domain. The range of / is the set of elements in B that are the image of something in A; that is {b e B : there exists ana £ A with / ( a ) = 6}. When f : A —¥ B has the additional property that f(x) — f(y) implies x = y, we say / is a one-to-one (1 — 1) function. Sometimes, rather than this definition we use its contrapositive which yields / is a 1 — 1 function if x 7^ y implies f(x) 7^ f(y)If / is 1 — 1, then there exists an inverse function f~l : B —> A defined by y = f~1{x)

if and only if x = f{y).

If, for any set C, we define / ( C ) as {f(x) : x 6 C} we say / is onto if /(A) = B. A function which is both 1 — 1 and onto is called a bijection. Bijections are mathematically important because if / is a bijection, then so is / _ 1 . We will use bijections when we discuss cardinality. The inverse function / _ 1 should not be confused with the inverse image of a set. If / : A —^ B and U C B, then the inverse image of U is the set of all values in A which map to something in U. Notationally, that is f-\U)

=

{xeA:f(x)eU}.

A function does not have to be a bijection or have an inverse in order to talk about inverse images. For example, if f(x) = x2 (with domain E; always assume the biggest domain possible), then / - 1 ([0,1]) = [—1,1]. Given / : A —>■ B and D C A, we say the restriction of f to D is the function / with the domain narrowed down to just the set D. This is written as f\p. These restrictions can have important consequences. The function F : R -> R given by F(x) = sin(x) is not a 1 — 1 function, but if D = [—7r/2, TT/2], then F\D is a 1 - 1 function. One special type of function we will use often is called a characteristic function. Given a set D C R, the value of the characteristic function of D (also called the

10

BASICS

indicator function) has values 1 for points in the set and 0 otherwise. Usually charac­ teristic functions are denoted using a lowercase Greek letter chi, x, for the function name with the set as a subscript. So this means for D XD{X) =

U

X{D.

Some functions have a special property involving the distance between two inputs versus the distance between their respective outputs. A function / : D —> R is called a Lip'schitz function if there exists a constant k > 0 so that for every x,y £ D \f(x)-f(y)\ 0 there exists a J V e N such that n > N implies \xn — x$\ < e. We can write this as lim xn = X(j. n—>oo

If a sequence does not converge, it is said to diverge. A sequence is called increasing (decreasing) if for any pair of natural numbers n and m, if n >TO,then xn > xm (xn < xm). If there are strict inequalities instead, then the sequence is strictly increasing (strictly decreasing). A sequence is called em monotone if it is either increasing or decreasing. Limits of a sequence brings us to limits of a function. The real number L is the limit of / as x approaches xo if for every e > 0 there is a S > 0 such that 0 < \x — xo\ < S implies |/(a;) — L\ < e. This means as the IT—values get closer to, but not equal to, x0 the /(a;)—values are approaching (near or equal to) L. The fact that the x's are not supposed to equal XQ is reflected in the requirement 0 < \x — XQ\. Also, notice that we are not requiring the limit to be the value of / at xo (or even that f(x0) exists). Another important property for a function is continuity. Intuitively we say / is continuous if we can draw its graph without lifting the pencil/pen/chalk/marker. We will need to be much more precise here. So we start with continuity at a point and then expand. This is the typical definition of continuity that one finds in a calculus text. It will be followed by the more precise 6-e definition. We will go beyond these when we study both topological and metric spaces.

REVIEW

11

Let / : A -*■ R be a function and x 0 an arbitrary value. We say / is continuous at XQ if the following three conditions are satisfied: 1. /(xo) exists, 2. lim^^^,, / ( x ) exists, 3. lim x ^ X o f(x) = f{x0). Continuity of a function / at the point x0 is equivalent to the statement that for any value e > 0 there exists a 8 > 0 so that for any x in the domain of / with |x — xo| < S we have |/(x) — /(xo)| < e. A function is called continuous if it is continuous at each point in its domain. If / is not continuous at some point, we refer to that point as a point of discontinuity for / . As an example of how 5 — e proofs are written, let us look at the following two examples, complete with scratch work. While scratch work is a necessary part of coming up with a proof, it should not be included in the proof. That said, many proofs are a matter of writing scratch work backwards, with some verbiage thrown in. fl

EXAMPLE 2.3 Show that lima;_>2(3x — 1) = 5. Scratchwork: If we look at \f(x) — L\ we have |(3x — 1) — 5| or |3x — 6|. This needs to be related t o | x - a | = | x - 2 | . Factoring gives us

|3x-6| = 3|x-2|. So if |3x — 6| = 3|x — 2| < e, then \x — 2| < e/3; and since we need to have |x — 2| < 5, this will be accomplished if 5 = e/3. Proof: Let e > 0 be given. For this e, let 5 = e/3. Then if 0 < \x - 2| < 5, we have |/(x) -L\ = |(3x - 1) - 5| = |3x - 6| = 3|x - 2| < 3 1/0*0 - f(x0)\

= |x 2 - 9| = |x - 3| • \x + 3|.

12

BASICS

This is good because \x — 3| is our \x — XQ\, but what to do with |x + 3|? Can we just divide both side by \x + 3|? No, although it seems every math major has at one point in his/her education. Instead we must replace \x + 3| with a constant. This is accomplished by making an assumption about \x — 3|. Assuming \x — 3| < 1 (the right side is arbitrary, but usually it is set to 1), this unfolds to — 1 < x — 3 < 1 or 2 < x + 3 < 4. So if \x — 3| < 1 we know \x + 3| < 4. Then |/(a;)-/(a:o)| = | s - 3 H z + 3 | < 4 | a : - 3 | . This takes us to the end of our scratch. If \x — 3| < d, then the right-hand side above is smaller than 46 and we just need to set 5 = e/4. It is vitally important to realize that in this proof we have two restrictions on \x — 3|. It must be less than 6 and less than 1 at the same time. Proof: Let e > 0 be given. For this e, let 5 = min{l, e/4}. Then if \x — 3| < 5, we have I/O) - Z(3)| = \x2 - 9| = \x - 3| • \x + 3| < \x - 3| • 4 < 45 = 4(e/4) = e. Since e was arbitrary, this holds for all values of e and thus the proof is complete.

4

On the real line, the distance between two points is given by d(x, y) = \y — x\. Different spaces may have different idea of distance. These distance functions are called metrics. The requirements for a function to be a metric are as follows: A metric on a set S is a function d : S x S —► [0, oo) which satisfies the following three properties: ■ For all x, y 6 S, d(x, y) > 0 (called nonnegativity) and d(x, y) = 0 if and only if x = y. • For all x,y £ S1, d(x, y) = d(y, x) (symmetry). ■ For all x,y,z S

G S, d(x, y) < d(x, z) + d(z, y) (The Triangle Inequality).

EXAMPLE 2.5 Suppose 5 is a set with three points in it, S = {pi,f2)P3}- We define for non-equal pi the distance to be d{pi,P2)

=d(p2,pi)

= 1,

d{p2,P3) = d(p3,p2)

= 2,

d(p3,Pi)

= 3.

=d(pi,p3)

Of course, d(pi, pi) — 0 for i = 1,2,3.

■

REVIEW

13

A space with a distance on it is referred to as a metric space usually written as the ordered pair (S,d). Every set can have a distance function defined on it as there is always the so-called trivial metric. 0, x = y, d(x,y) 1,

x^y.

Although these definitions and concepts in this chapter are written for the real line, we will explore various generalizations. These include going from R to R", looking at differently structured spaces like metric and topological spaces, and gen­ eralizing our ideas to appropriate types of limits or continuity.

14

BASICS

EXERCISES 2.1

Let A = {1, 3, 5, 7,9} and B = {1, 2, 3,4} in the universal set U = {1,2,3,4,5,6,7,8,9,10}.

Lwf f/ze members of a) AuB b) AC\B c) Ac d) A\B e) Ax B

f) By. A 8) (B\Af 2.2

Explain why for any set A we have ^ 1 x 0 = 0 x ^ 1 = 0.

2.3

Let An = {1, 2 , 3 , . . . , n} for n = 1,2,3,.... Determine UnAn and C\n An.

2.4 G/ve an example of a non-empty, closed set in M. whose boundary consists of a single point. 2.5 set.

Give an example of a non-empty, open set in R whose boundary is the empty

2.6 Determine whether or not the given sequence converges. A proof is not neces­ sary here. a) •&TI = (3n2 - 5n)/(2r7,2 + 3) b) Vn = l + ( - l ) n c) an = sin(n7r/2) d) r ■■= (1 + 1/n)" e) $n '= (y/n + 1 - v/n) n

2.7

Show that [0,1] is a Qs set.

2.8

Prove that f(x) = x3 is continuous at XQ — 1.

2.9

Prove, using an e — N argument, that xn = 1/n converges to 0.

2.10

Prove, using an e — N argument, that yn = sin(n)/n converges.

2.11

Show that f(x) = -^rh is a 1-1 function, but not onto R.

2.12 Explain why f(x) = sin(x) is a Lipschitzfunction. (Hint: This uses the Mean Value Theorem from Calculus.) 2.13 Show that any linear function, f(x) = rax + b, m ^ 0 must be a Lipschitz function.

EXERCISES

2.14

15

Consider the function

jxsin(l/x),

x ^ 0,

a) Show that ifa = 0, then f is continuous at x = 0. b) Prove that for any other value of a f is discontinuous at x = 0. 2.15 Let I be an interval in K.. A function f : I —> R is called uniformly contin­ uous iffor every e > 0 there exists a S > 0 such that ifl x,y £ I with \x — y\ < S, then \f{x)-f{y)\<e. (compare this with the s — 5 definition for ordinary continuity). Show that any linear function f(x) = mx + i, m / 0 must be uniformly continuous. 2.16 Prove that f : [1, oo) —> K given by f(x) = y/x is a uniformly continuous function. 2.17

Prove that any Lipschitz function must be a uniformly continuous function.

2.18

Show that Example 2.5 satisfies the requirement of being a metric.

CHAPTER 3

INTRODUCING THE CANTOR SET

Why is the Cantor Set interesting? Well, we hope to find answers to that question as we continue to read this book. One thing we can say is that the set seems to pop up everywhere. When looking for an example to illustrate a definition some kind of Cantor Set usually does the trick. This can happen in real analysis, topology, abstract algebra, probability theory, fractal geometry, the list go on. In searching for extreme examples, many times one begins, "Start with a Cantor Set and ...." It is this utility of the Cantor Set which makes thorough knowledge of it an important item in a mathematical toolbelt.

3.1

Some Definitions and Basics

There are several ways to create a Cantor Ternary1 Set and we shall first give the standard or canonical way of making one. Start with the closed unit interval (K0 = 'Ternary refers to the fact that we will divide things into thirds. For the most part, we will drop this word and leave it to the reader to understand when we mean this set and when we mean some general Cantor Set. The Elements of Cantor Sets -With Applications, First Edition. By Robert W. Vallin Copyright © 2013 John Wiley & Sons, Inc.

17

18

INTRODUCING THE CANTOR SET

[0,1]). Divide it into three equal sections and remove the open middle third. Thus we now have K\ = [0,1/3] U [2/3,1]. We then continue inductively. At step n, Kn consists of T1 closed subintervals. For the (n + l)st step, divide each of the closed subintervals in the previous step into thirds and remove the open middle third. Continuing indefinitely gives us the collection of sets {Kn}^L0. Finally, the Cantor Set is given by oo

ci/3 - n K n A representation of the first four steps is shown below: K0= Kx= K 2= K3 = Notice that since we are only removing from the middle of an interval, any point that is an endpoint of a interval in Kn will be in C1/3, however, there are many more points than just these. Is there something magical about dividing things into thirds? No, of course not. One way to generalize this is to change the ratio of the interval removed. At each step in this process we have an interval I and from that interval we remove a middle subinterval J so that £(J)/£(I) — 1/3, where £{■) refers to the length of an interval. We can create a new Cantor Set by insisting the middle interval removed have ratio £{,])/£{I) = r where 0 < r < 1. We will denote this Cantor Set by Cr. Other generalizations of the Cantor Set will be considered in Chapter 10. Here is an easy first question. Since we start with the closed unit interval, we are beginning with something of length one. Then we take out the open interval ( 1 / 3 , 2/3) leaving us with two closed subintervals each with length 1/3. From those, we remove one interval each and each of those subintervals has length 1/9. Now we have four closed subintervals of length 1/9 each. So how much is taken away in total? Theorem 3.1 The total length of the subintervals removed in the derivation 0/C1/3 is one. Proof: At each stage n > 0 there are 2n~1 open subintervals removed and each of these subintervals has length (1/3)™. The total length removed is then represented by the infinite geometric series Yl^Li 2 " ~ 1 ^r. From Calculus we know if \r\ < 1, then 5Z^Lo a ■ rn = a / ( l — r ) . Thus 0 0

]T 2"" n—\

1

1

- = 1/3 + 2(1/9) + 4(1/27) + • • • =

1/0 T

^ —

= 1.*

'

So in a way, we start with an interval that has length 1 and remove from it intervals whose aggregate length is also 1, yet there are plenty of fascinating points leftover.

SOME DEFINITIONS AND BASICS

19

Let us refine our characterization ofpoints in (or out of) the Cantor Set. Normally we write numbers in base 10. With base 10 we use the digits 0,1, 2, 3 , . . . , 9 and the position of a digit refers to a power of 10. For example, in 123 the digit 2 means 2 ■ 102 or 200, where for 2103 the 2 means 2 ■ 103 = 2000. While the symbol remains the same, the meaning changes with its position and is interpreted as a multiple of a power of 10. Similarly, a number to the right of a decimal point refers to negative powers of ten; for the number 0.58 the 5 refers to 10 _ 1 while the 8 is for 10~2. If we let b be an integer greater than 1, we can write numbers in base b, using powers ofb rather than 10 and b digits 0 through b. In base 5 the digits are 0, 1, 2, 3, 4; In base 2 (binary) the digits are 0, 1; in base 12 (duodecimal) we use, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, and E. We will use the subscript b to say a number is written in that base. So 324s refers to base 5. As for going back and forth between bases b and 10, converting base b to base 10 is done by multiplying out the powers of the base; so 3245 = 3 x 5 2 + 2 x 5 1 + 4 x 5 ° = 75 + 10 + 4 or 89 in base 10. Converting from base 10 into another base can be done by finding the most copies ofbn that "fit" into the numberfor each positive integer power ofn. The number 278 can be written as 2x125+1x25+3x1 = 2 x 5 3 + l x 5 2 + 0 x 5 1 + 3 x 5 ° = (2103)5. A second way to do this is with a division trick, looking at the remainders from repeated division (by 5 in this case since we 're converting into base 5). 278 -T- 5 = 55 with a remainder of 3, 55 4- 5 = 11 with a remainder ofO, 11 -f- 5 = 2 with a remainder of 1, 2 + 5 = 0 with a remainder of 2. We stop here since the dividend is 0. Now if we read the remainders bottom to top we get 2 - 1 - 0 - 3 which tells us 278 = (2103)5. To convert a base 10 decimal into another base, b, we multiply repeatedly by b and collect whole number parts. For example, turning 1/5 = 0.2 into a base 4 number we have 0.2 x 4 which is 0.8 whose whole number part is zero. This means the first spot to the right of the base 4 quaternary point is a zero (of course we cannot say the "tenths place" as we are not in base 10). Then 0.8 x 4 has whole number part is 3 which places a 3 two spots to the right of the decimal. After removing the 3 there is 0.2 leftover and the cycle starts all over again. Thus - = (0.030303...) 4 . o

For a second example of a rational number expansion we look at the number 35/9 and a second technique for computing the base 7 number. This method uses factoring.

20

fl

INTRODUCING THE CANTOR SET

EXAMPLE 3.1 In this method we are factoring out powers of 7 along with converting the nu­ merator into a sum where the first time is a multiple of the denominator 9. 35 9

_

27+8 _ Q _ L 8 _ O , 1 5 6 _ O , 9 9 79

=

3 + y + jz + Yi ( g ) = 3 +

1 54+2 7 9

y + if2 + 73 (-9-) ,

at which point our sequence repeats itself. Thus we arrive at — = (3.613613613... ) 7 .

The construction of a Cantor Set fits right into base 3. When we divide the unit interval into thirds, we can look at this as splitting the numbers up according to their first digits. The left section consists of numbers whose first decimal digit (in base 3) is a zero, the middle section has all numbers whose first decimal is 1, and the right section has numbers whose first digit is 2. Removing the open middle rids us of all numbers whose first digit is a 1. When the left interval is subdivided into thirds and the middle removed, what we are keeping is all the numbers between 0 and 1 which begin in base 3 as .00 and .02. Technically, we have to be a little careful as some numbers are not unique in their representation. For example, in base 10 1.999999... = 2.0000.... Now if we write our numbers between 0 and 1 in base 3, we can say the Cantor Set consists of all numbers which have a base 3 representation consisting only of 0 's and 2 's to the right of the ternary point and a 0 to the left of the ternary point. The endpoints of the intervals in the creation of the Kn, such as 1/3 or 8/9, are points in the Cantor Set and correspond to those numbers in base 3 can be written that way. Quick arithmetic shows

jj = (0.22000... ) 3 and i = (0.1000...) 3 = (0.0222... ) 3 . Both of those are endpoints of intervals in some Kn. There are points in C\/% that are not endpoints of intervals (we can tell a point is at the end of some interval because it will have a finite expansion in base 3). For an example of a non-endpoint in base 3 we have (0.02020202...) 3 , which is 1/4. Let us say a bit more about what goes on with points in the Cantor Set. This comes from [53] where the rational values in C are partitioned.

SIZE OF A CANTOR SET

21

The numbers that are rational in base 3 are the numbers that can be written with finitely many 2 's after the ternary point (the name for the decimal point in base 3). These numbers look like x\

x2

x3

3

2

3

3

3

t

xn

xi3n_1+x23n~2 +

3"

V xn

3"

where each Xi is either a 0 or a 2. These correspond to the left endpoints of the removed intervals in the Cantor Set construction. Purely periodic points have the form Q,(x\X2Xz ... xn)s again with Xi G {0, 2}. Expanding these lead us to x^-1

+ x23n~2

+ ■ ■ ■ + xn (

On

\

, 1 , Qn

1

g2ra

and cleaning up the geometric series gives us xx?,n-1

+ x 2 3"~ 2 + • ■ • + xn 3™-l

Numbers in this category include 1/13, and 1/10. The rest of the rational points are mixed rationals, those whose repeating patterns do not start after the ternary point, but instead become periodic after some terminat­ ing number. For example, (0.220)3. We can find this number's value in base 10 by relating it back to a purely periodic number. Here we see (0.220)3 = (0.2)3 + (0.020)3 = (o.2 + ^ 0 . 2 o ) 3.2

= | + \ ( ^

= ^.

Size of a Cantor Set

The size of a Cantor Set is an important question. Is it a big set or small? Before we can answer this question, we must define what we mean by saying a set is small. This notion comes from Van Rooij and Schikhoff [75]. We say a collection of sets S from K is a collection of small sets if 1. R ^ <S (the whole line is not small). 2. If'A € S and B C A, then B G S (a subset of a small set is small). 3. If A is small, then for every fixed value b so is b + A = {b + a : a G ^4} (the translation of a small set is small). 4. IfAn, n > 1 are each small, then so is UAn (the countable union of small sets is small). Some of the collections we shall look at that qualify as a small set include countable sets, measure zero sets, and first category sets. It is worth noting that finite sets do not meet this idea of a small set. As we shall see, in some ways the Cantor Set is small and in other ways large.

22

INTRODUCING THE CANTOR SET

3.2.1

Cardinality

Cardinality deals with counting the number of elements in a set. Cantor's work in this area opened mathematics up to the ideas of what it means for a set to be infinite and the notion of different types of infinity. First we need to recall two properties a function from Chapter 2, followed by examples. Definition 3.1 Let A and B be sets and suppose f is a function with domain A and range B. We say f is one-to-one iffor a ^ b in A we have f(a) ^ f(b). The equivalent definition one sometimes sees is f(a) = f(b) implies a = b. We say f is onto B if for every b G B there exists an a E A such that f(a) = b. If f : A —> B is one-to-one and onto, then we say f is a bijection.

fl

EXAMPLE 3.2 Let f : R -» K be defined by 1. f(x) = ex. Then f is one-to-one. but not onto. 2. f(x) = x3 — x. Then f is onto, but not one-to-one. 3. f(x) = 2x + 4. Then f is a bijection. u

Now we are ready to look at equivalence between sets and cardinality. As we will see, cardinality, even if not known under that name, is something with which the reader is quite familiar. Definition 3.2 Let A and B be two sets. We say A is equivalent to B if there exists a bijection, f, between A and B. Our notation for equivalent sets is A = B. For example, A = {a, b, c} and B = {x, y, z} are equivalent while there is no equivalence between A and C = {a, /3,7,5}. Our equivalence = is a special instance of an equivalence relation on a collection S. Let us take a step back and define equivalence relations and partitions before we move on. Definition 3.3 Let S be a set. A relation , ~, on S is a subset of S x S. We write

to say, "x is related to y."

SIZE OF A CANTOR SET

fl

23

EXAMPLE 3.3 1. On the set Z say x ~ i y if and only ify — x is an even number. 2. On the set S = {x : x is a person with the first name Sam} we let x ~2 y if and only if x and y are siblings. 3. On the set K we say i ~ 3 j / if and only ifx < y. u

The reflexive, symmetric, and transitive properties are important, especially when a relation ~ satisfies all three at the same time. Definition 3.4 Let ~ be a relation on the set S. We say the relation is 1. reflexive if for every x G S it is true that x ~ x. 2. symmetric if for every x,yESifx~y, 3. transitive if for all x,y,z

then y ~ x.

G Sifwe know x ~ y and y ~ z, then x ~ z.

If a relation is reflexive, symmetric, and transitive, then it is referred to as an equiv­ alence relation. fl

EXAMPLE 3.4 For the relations above, ~ i is an equivalence relation, ~2 '•$ only symmetric, ■ and ~3 reflexive and transitive, but not symmetric.

When the set S has an equivalence relation on it, we can collect equivalent ele­ ments together. These collections are called equivalence classes. Definition 3.5 Let S be a set and ~ an equivalence relation on S. For any x G S we define the equivalence class of x as

\A = {y e s: y ~ x}. [ 8 EXAMPLE 3.5 Let S be the set of cards in a standard deck of 52. For two cards x and y, we say x is related to y if they are both diamonds (or both spaces, both hearts, both clubs). This is an equivalence relation and the equivalence classes are the suits clubs, spades, hearts, and diamonds. If instead x is related to y by them having the same value, then there would be 13 equivalence classes: aces, deuces, treys, ..., queens, and kings. ■

24

S

INTRODUCING THE CANTOR SET

EXAMPLE 3.6 Let P be the set of all polynomials with integer coefficients. We say p(x) is related to q(x) if they have the same y-intercept. This is an equivalence relation and the equivalence classes are the functions which all agree at 0. Using our notation, for a function p(x) we have [p(x)] =

{q(x)eP:q(0)=p(0)}.

Furthermore, if p{x) = 3x2 + 2x + 5 we could see that q(x) = —7x + 5 is a member of \p(x)] while r(x) = x3 — x1 + 4 is not. I The theorem below gives us three important properties for equivalence classes. Casually, these say equivalence classes are like the pieces of a jigsaw puzzle. Each piece contains part of the picture, no two pieces contain the same part, and together they make up the whole scene. Theorem 3.2 Let S be a set and ~ an equivalence relation on S. Then the following are true for the equivalence classes [x] induced by ~. 1. For each x £ S, [x] ^ 0. 2. For each x,y £ S either [x] = [y] or [x] n [y] = 0. 3. UxeS[x] = S. Proof: We will prove part 1 only. Parts 2 and 3 will be left for homework. Pick x £ S and look at [x\. Since ~ is reflexive, x ~ x which means x G {y € S : y ~ x} = [x\. Therefore [x] ^ 0. 4 Now let us get back to cardinality. We say sets A and B are equivalent if and only if there exists a bijection f : A —> B. It is fairly obvious that any set is equivalent to itself so we have that = is reflexive. Any bijection f has an inverse / _ 1 which is also a bijection defined by f~l{b) = a if and only if f(a) = b. Thus the symmetric property is satisfied. Finally, if f is a bijection between A and B and g is a bijection between B and C, then the composition go f is a bijection between A and C. Hence = is transitive and we have an equivalence relation. This means we have equivalence classes and that is where we willfindcardinality. Definition 3.6 Let A be a set. If there exists a natural number k such that A = { 1 , 2 , 3 , . . . , k}, then A £ [{1,2,3,..., k}] and we say A is a finite set with cardi­ nality n(A) = k. Otherwise A is an infinite set. So we have a strict definition to show that A — {a, b, c] is a finite set with cardi­ nality 3. Definition 3.7 A set A is countable if it is either finite or equivalent to N. When the latter is true the set is called countably infinite. If a set is not countable, then it is called uncountable.

SIZE OF A CANTOR SET

25

When a set in M. is infinite, its cardinality is denoted using the Hebrew letter aleph (KJ. For countably infinite we use Ho. The collection of countable sets are our first type of small set using the criteria of [75]. Let us look now at examples of countable and uncountable sets and the proofs of their various cardinalities. But before our first result, let us state a lemma. Its proof is left as a homework exercise, but we shall use it right away in Theorem 3.3. Lemma 3.1 An infinite subset of a countable set is countable. Theorem 3.3 The set Q+ = {x\x is a positive rational number} Z + , q / 0 } i s a countably infinite set.

= {p/q\p,q

Proof: We begin by writing out the positive rationals in the following 1

2

3

4

5

■••

1/2

2/2

3/2

4/2

5/2

•■•

1/3

2/3

3/3

4/3

5/3

•••

1/4

2/4

3/4

4/4

5/4

•■•

G

way:

Starting the upper leftmost corner and moving diagonally left side up to top, we get Y

2^

3*

jP"

^

P?2

2/2

3/2

4/2

5/2

•••

#3

2/3

3/3

4/3

5/3

••■

yf4

$fA

3/4

4/4

5/4

•••

This gives us the one-to-one 1

1/2

1 2

■■■

matchup 2

1/3

2/2

3

1/4

2/3

■••

3

4

5

6

7

8

••■

Thus the first array is equivalent to N. Technically there are repeats (after all 1/2 = 2/4 and 3/1 = 6/2), but regardless, this shows the positive rationals are equivalent to the natural numbers and hence have the same cardinality. 4 Theorem 3.4 If each set An, n > 1 is a countable set, then the union U ^ L ^ n is countable. Proof: Similar to the proof for the rational numbers, each Ai can be written out as M = {ai,i,a1]2,ai,3,---},

26

INTRODUCING THE CANTOR SET

A2 = {a2,l,02,2,02,3,•■■}, A3 = { a 3 , i , a 3 , 2 , a 3 i 3 , . . . } , Ai = {0.4,1,04,2, 04,3, • • •},

where a,ij refers to the j element in the i set. Then a diagonal argument puts these element m l — 1 correspondence with N. 4(t From dealing with functions, we are used to ordered pairs (a, b). In general, given two sets A and B we define Ax

B = {(a,b) : a e Aandb

e

B}.

There is nothing special about having only two sets, so this can be fully from an ordered pair to an n—tuple by the following.

generalized

Definition 3.8 Let Si, i = 1, 2 , . . . , n be a collection of sets. We define the Cartesian product of the Si by Si x ^2 x ■•• x Sn = {{x1,x2,...,xn) Each element {x\, x2 ■ ■ ■ ■, xn) is called an (ordered)

: xt G Si}. n-tuple.

Theorem 3.5 IfSk is a countable set, k = 1, 2 , 3 , . . . , n, then Si x 5 2 x S3 x • ■ ■ x S n is also a countable set. Proof: This is yet another example of a diagonal argument and will be left as an exercise. 4k This extends to countably infinite many countable sets. Theorem 3.6 If Sk is a countable set, k = 1, 2, 3 , . . . , then Si x S2 x S3 x ■ • •

is also a countable set. In 1874 Cantor published a proof that an interval of the form [a, b] is uncountable. Later, in 1891, he published another paper containing his famous "diagonalization" proof of this. We give our own version of the proof now. Theorem 3.7 The unit interval [0,1] is uncountable. Proof: Every number in the unit interval can be written as 0.did2ds ..., where each di is a digit from 0 to 9 and (for sake of having the representation be well-defined)

SIZE OF A CANTOR SET

27

no number ends in an infinite string ofO's, but instead as an infinite string of 9's; for example, 0.135000 . . . will be written as 0.1349999 . . . . This is the only possible redundancy in representations. Proceeding with a proof by contradiction, suppose the numbers in [0,1] are countable. Then we can put the set [0,1] into 1 — 1 correspondence with N as follows: a\ = Q.d\d\d\ . . . ,

a2 = O.dldldl..., a3 = O.dldldl...,

where d\ refers to the ith digit in the decimal expansion of the number a,j. Let us create a number x G [0,1] by x = O.&16263 • • • where bi = 3 if ai 7^ 3 and bi = Aifai — 3. Then x ^ a,j for any j since the two numbers must disagree in the jth spot. This contradicts our list being a complete roster of the numbers in [0,1]. Thus the unit interval is uncountable. 4 The cardinality of the real numbers is 2N°. We note, without proof, that the cardi­ nality of the unit interval is also 2^° Corollary 3.7.1 The set of irrational numbers in [0,1] is uncountable. Proof: We do this by contradiction. Suppose the irrationals, (K \ Q) n [0,1], were a countable set. Then since the union of countable sets is again countable, we have

[o,i] = (Qn[o,i])u((R\Q)n[o,i]) is a countable set, which we know is false, 4k The next result, which we include for completeness, will not be followed with proof. It comes from the fact that if A is uncountable and A C B, then B is uncount­ able. Similarly a subset of a countable set must be countable. Corollary 3.7.2 The set R is uncountable. For an example of something different, we look at another way (other than ra­ tional/irrational) to partition the real numbers. We say x is a algebraic number if it is the root of a polynomial with integer coefficients. So \f2 is algebraic as it is a solution to x2 - 2 = 0. If a number is not algebraic, then it is transcendental. Well-known transcendental numbers include ir and e. There are many, many others. The proofs for n and e being transcendental are non-trivial and will not be presented here.

28

INTRODUCING THE CANTOR SET

Theorem 3.8 The set of algebraic numbers is a countable set. Proof: Each algebraic number a is associated with a polynomial p(x) with integer coefficients. So we will show this collection of polynomials is countable. Let us fix n 6 N . Then every polynomial of degree n with integer coefficients can be matched up with an (n + l)-tuple in the following manner: p{x) = c„x n +c„_ 1 x n _ 1 +c„_2^"~ 2 H

\-CiX+c0 ^

(c„,c„_i,cn_2,...,ci,c0).

These (n + \)-tuples are countable since they consist of integers. Next, the union over all values of n of these (n + \)-tuples is countable and taking this backwards to the polynomials with integer coefficients they are countable. Finally this brings us that the set of algebraic numbers is countable. 4k Corollary 3.8.1 The set of transcendental numbers is uncountable. Proof: This is the same as the proof for the irrational numbers. 4t The Cantor Set is an uncountable set as we shall prove. This really shows the paradoxical nature of the set. We start with the unit interval, [0,1], remove open intervals (themselves uncountable) whose total length is one, yet what is leftover has the exact same cardinality as the unit interval. Theorem 3.9 The cardinality of the Cantor Set is 2N°. Proof: As we have already seen, the Cantor Set can be represented by the set of all numbers between 0 and 1 which have a base 3 representation that consists of only 0's and 2's. We construct a function from C1/3 to [0,1] by using the base 2 (binary) representation of number in the unit interval. Given a number in the Cantor Set we can think of it as x = 0.ai 0 there exists an N G N such that for n > N \xn ~ %o\ < e. However, in a topological space there is not necessarily an idea for "distance" to say closer. Our convergence in R is equivalent to saying that for any positive e there is a natural number N so that ifn is greater than N, xn is in (XQ — e, Xo + s). Since some K the tail of a sequence, this becomes for every e > Ofor we call {xn}n>jifor K large enough all the tails are in the open interval (XQ — e, xo + e). Open intervals are what we generalized to get to open sets in a topological space. This leads us to this definition of convergence. Definition 3.12 Let (S, T) be a topological space and {xn} a sequence of elements from S. Then we say {xn} converges to XQ 6 S if for every set U G T such that xo £ U there is an N G N such that ifn > N, then xn e U. As the examples below show us, strange things can happen depending on the topology on the space. Before the first example we need a definition. We say a sequence is eventually constant if, for some c s S, xn = cfor n> N.

SIZE OF A CANTOR SET

fl

31

EXAMPLE 3.8 Let S have the discrete topology. If{xn} is not eventually constant, then {XQ} is an open set which does not contain any tail of the sequence. Thus the only convergent sequences are the eventually constant ones. ■

S

EXAMPLE 3.9 For S with the trivial topology, every sequence converges to every point. Let {xn} be a sequence in S at XQ an arbitrary point in S. The only open set containing XQ is S and xn € S for all n > 1. Thus since XQ is arbitrary, {xn} converges to any point. ■

For another example of how things extend from R to topological spaces, let us look at the idea of continuity. There is the 5 — e definition of continuity: A function / : R -> 1 is continuous at the point XQ if for every e > 0 there exists a S > 0 such that for all x with \x — XQ\ < 5 we have \f(x) — /(xo)| < £• As with sequences, this depends on absolute value (distance) in R which we do not necessarily have in a topological space. Looking closely, though, we can see that unfolding the absolute values shows that what we are dealing with are open intervals in M. which we can then generalize to open sets. To do so, we need to know about the inverse image of a set. So let f : S —>• V and U C V. The inverse image ofU is the set of all x € S which map into U. The notation for this is

r1(U) =

{xeS-.f(x)eU}.

Although the notation is the same, this is not the inverse function. We are not claim­ ing f is one-to-one so the inverse function may not exist. Definition 3.13 Let (S,Ti) and (V, Ti) be topological spaces, f : S —>■ V, and XQ 6 S. We say that f is continuous at x0 if for every open set U 6 7i with f(xo) € U the inverse image ofU

f-1(U) =

{xeS:f(x)eU}

is open in T\. In addition, we say f is a continuous function if it is continuous at each point in its domain. The example below shows how this works and some of the strangeness involved. fl

EXAMPLE 3.10 Let S be the set R given the Euclidean topology and V beM. given the discrete topology. Let f be the function f(x) = x (the identity function). If the domain is S and range V, then f is not a continuous function: {xo} is open in V, but not open in S. If things reverse themselves and the domain is V and range S, then not only is f continuous, but every function is continuous. m

32

INTRODUCING THE CANTOR SET

Denseness and category are defined for any topological space (S, T ) now. To get the full power of our big result (The Baire Category Theorem) we need the additional property of a space being complete. We are saving that particular property for the chapter on Self-Similar Sets. So in a bit we will switch to only working in M with the usual, Euclidean, topology. Definition 3.14 Let (S, T) be a topological space and let A be a subset of S. We say that A is dense in S if for every U € T there is an a G A such that a £ U. That is, every open set contains at least one point in A. Now we define an idea for a sparse type of set. We are trying to make things so that no part of the set can be thought of as dense. Definition 3.15 Let (S, T ) be a topological space and B a subset of S. Then B is nowhere dense in S iffor every U C S there exists an open setV .o+ "■ S° let e > 0 be given. Since n > 1 is fixed, for each Xj G E, de­ fine Oi = (x,i — e/2n,Xi + e/2n). Each Oi has diameter e/n and E C UOj.

SIZE OF A CANTOR SET

Now

37

n

y ^ £{Oi) = n ■ s/n = e. i=l

This means m* (E) < e and this gives m* (E) = 0. 4 Of course, our focus is on the Cantor Set, and since the total lengths removed is 1 it should be no surprise that our next example is to show m*(Ci/3) exists and, in fact, is zero. fl

EXAMPLE 3.13 The Cantor Set has outer measure zero.

u

Proof: Let e > 0 be given. Recall that at stage n in the construction Kn consists of2n disjoint closed intervals, I\, I 2 , . . . , hn each of whose length is 3~™. Each Ij then can be contained in an open interval Oj where £{Oj) < 3 _ n + 1 . The {Oj} are then a collection of open interval which contain Kn (hence also C1/3) and '2

^ £ ( 0 J ) = 2"-3-™+1 = 3 ( 1 So by picking n large enough we guarantee

m*(C71/3) 0, this means "1 = oo." Thus

Once we have some measurable sets, we can create more by combining them in the usual way. We show this in the theorem below. Theorem 3.15 Let E and F be measurable sets. The the sets Ec, EUF, and ECiF are all measurable sets. Proof: We will prove the result for E U F and leave the others as exercises. Let A C M. As noted above, we really only have to prove m(A) > m(An(EU

F)) + rn{An(EU

F)c).

The rules for DeMorgan 's Laws give us

A = (AnE)u(AnEc) = (AnE)u(AnEc nF)u(AnEc n Fc). Then the subadditivity of measure leads us to m(A)

=

m((A n E) U {A n EG n F) U (A n Ec n Fc))

>

m((A n E)) + m((A n Ec n F)) + m((A n Ec n Fc))

and we have E U F is a measurable set. 4 We also get the following corollaries which will end (for now) our look at mea­ surable sets. Note that these results tell us that measurable sets form a a-algebra. Corollary 3.15.1 Let {En} be a sequence of measurable sets. Then both UnEn and nnEn are measurable sets. Corollary 3.15.2 Every open set and closed set in M. is measurable. Let us end this part of the discussion with an application of measurable sets. As we know from calculus, if a function f is nice, we can find its Riemann integral. In the beginning we have continuous and positive functions for "nice" and introduce this integral to solve The Area Problem. Then our ideas are generalized so that functions neither need to be continuous nor positive-valued. So the question becomes, "What

40

INTRODUCING THE CANTOR SET

types offunctions are Riemann integrable? " A partial answer is the theorem below. This is presented without proof. Theorem 3.16 A bounded function f on an interval [a, b] is Riemann integrable if and only if the set of points at which f is discontinuous is a set of measure zero. Our eventual goal for the end of this section is to introduce Legesbue's integra­ tion. In order to get there, we shall now move on to the idea of measurable functions. Looking at these requires us to use some special functions so we will start with defin­ ing the characteristic function of a set. Characteristic functions are a big part of Lebesgue integration. Definition 3.19 Let Abe a set in R (in general, A is a subset of some abstract space S, but in this section we are just working in the real line). The characteristic function of A, written \A using the Greek letter chi, is given by XAix)

=

(l,

ieA,

\o,

X{A.

Definition 3.20 Let f be a real-valued function with domain E, a measurable set. We say f is a measurable function if and only if for every r G R, the set {xeE:f(x) 1). An unproved by us corollary of this is that any continuous function is measurable. This is a consequence of the following theorem. Theorem 3.17 Let E be a measurable set and let f : E —> K. Then the following are equivalent: 1. For every r G R the set {x G E : f(x) < r} = / _ 1 ((—oo, r)) is measurable. 2. For every r G R the set {x G E : f(x) < r} = f~1((—oo,

r]) is measurable.

3. For every r G R f/ie set {x G E : f(x) > r} = / _ 1 ( ( r , oo)) is measurable. 4. For every r £ R the set {x G E : f(x) > r} = / _ 1 ( [ r , oo)) is measurable. Proof: We will show the first implication. The rest comes from complements. Assume {x G E : f(x) < r} is a measurable set for any r. Then {x G E : f(x) < r + 1/n} is measurable for each n G N. Since measurable is closed under intersection {x G E : f(x) < r} = Dn{x G E : f{x) 0 there exists a 5 > 0 such that <e,

42

INTRODUCING THE CANTOR SET

for every Riemann sum associated with f on I with respect to P where mesh(P) S. If f is Riemann integrable on I, we denote R j


fc(/)A(Ifc), fc=l n

[/(/,P) = ^Affc(/)A(/fc). fe=i

These upper and lower sums use the area under step functions. A step function with domain [a, b] is made by partitioning the interval into a = to < t\ < ti < ■ ■ ■ < < tn = b and having f(x) = Ci, a constant on each subinterval, (i,;_i,t,). tn-i In these cases the supremum and infimum of f on the subinterval form the constant values. Definition 3.24 Let f be defined on I. Then we define the Upper and Lower Dar­ boux integrals of f by [ / ( / ) = inf { [ / ( / , P) : P is a partition of I}, L(f) tfU(f)

= s u p { L ( / , P) : P is a partition of I}.

— -^(/)> then we say f is Darboux integrable on I and denote this common

value as R J

f.

It is a fairly straighforward Advanced Calculus exercise to prove that the Riemann and the Darboux integrals are the same. Although the Riemann definition is typically what one sees in a Calculus course, the example below will show us the usefulness of the Darboux definition. Notice that there are some restrictions in these definitions. The interval of inte­ gration must be closed. The function f has to be bounded. These two conditions, while necessary, are not sufficient to make a function integrable.

SIZE OF A CANTOR SET

fl

43

EXAMPLE 3.15 Let A = Q n [0,1]. Then the characteristic function

XA[X)

fi, \o,

ieQn[fl,i], ^Qn[o,i]

« nof a Riemann integrable function. On any subinterval Ik we have Mk{f)

= landmk(f)

= 0.

Thus for every partition P n

n

L(f, P) = 52 mfc(/)A(Jfc) = ] T 0 ' A ( ^ ) = ° fc=i fc=i

and n

n

fc=i

*;=i

u(f, p) = J2 Mk(m(h) = E 1 ■ A ( 7 *) = ^ giving us 0 = L ( / ) < £/(/) = 1, hence f is not integrable.

■

Lebesgue integration uses a special type offunction to approximate the measur­ able function f called a simple function. These are basically sums of characteristic functions. Definition 3.25 The real-valued function g is called simple if it assumes only finitely many values a\, 02, ■ • •, an. If a function is simple, then it must have the form 9 = ^2ai

-XAi,

where Ai = {x : g{x) = a{\. For our use, we will insist that each Ai be a measur­ able set. This is called the canonical representation of g. There are other ways to represent g. For example, if g is 1 on A = [0, 2), we could also say it is ] on AiD A2 where A\ = [0,1) and Ai = [1, 2). Although this seems unusual, we will have need of this in proving theorems. S

EXAMPLE 3.16 A step function, like 2,

m = { 1, 10,

ifx > 3, if l < x < 3, ifx < 1

44

INTRODUCING THE CANTOR SET

is an example of a simple function.

■

However, not every simple function is a step function. We see this in the next example. fl

EXAMPLE 3.17 Let A be any set. Then \A {x) is a simple function. Consider A to be the integers. Then this simple function is not a step function. ■

// g is a simple function that is nonzero on a set of finite measure, then we can define the integral of g. Before we begin, recall that a collection of sets {Ai} are pairwise disjoint if for all i ^ j Ai n Aj = 0. Definition 3.26 Suppose g is a simple function which is 0 outside of a set E with finite measure. Furthermore say g(x) = Yl't-i ai' XAt {%) where the Ai are pairwise disjoint, measurable, and UAi = E. Then we define the (Lebesgue) integral of g to be P

n

I 9 = V] alm(Al

JE

The usual results regarding sums and multiplying by constants holdfor integrating these simple functions. Before we prove it, though, we need this lemma. Lemma 3.3 Let g = YH=i aiXE% where we have Ei are pairwise disjoint. If each Ei is measurable and m(Ei) < oo, then n

/

9 = ^2alm(El)

Proof: This is a consequence of two facts: (I) Measure is subadditive, and (2) Ai = Ua=aiEi. Then a ■ m(Ai) — ^

a, • m(Ei).+

ai=a

Theorem 3.18 If g and h are both simple functions which are nonzero on sets with finite measure, then

(ag + Ph) =a

g+ 0

h,

where a and /3 are real numbers. Proof: Suppose that {Gi} and {Hi} are the sets from the canonical representations of g and h, respectively. Define the finite collection of sets Ei by taking all the

SIZE OF A CANTOR SET

45

possible intersections Gi fl Hj. These Ek are measurable and pairwise disjoint with the additional property that

9 = YLakXE*' and

n h =

Yl hkXEk ■ fe=l

This gives us

{ag + ph) = Y^iaak + 0bk)XEkk=i

Then it is an application of the Lemma that j {ag + j3h) = ^T(aak

+ 0bk)mEk

=a

g+0

h.

Corollary 3.18.1 Iffor simple functions g and h, g > h almost everywhere on E, a set with finite measure, then J g > J h. Proof: This proof is left as a homework exercise. We are now ready to define a Lebesgue integral for a bounded function f. We do this by approximating with simple functions similar to the way step functions are used to approximate the Riemann integral. Definition 3.27 Let f be a bounded, measurable function defined on a measurable set E, where the measure of E is finite. Then the Lebesgue integral of f over E is defined by

f f = mf / 5,

IE JE

J^9JE f the characteristic function of the rational numbers. This function is has upper sum U(f, P) — lfor any partition P and lower sum L(f, P) = Ofor any P. Thus U(f) ^ L(f) and so it is not Riemann integrable. On the other hand, f is constant at 1 except on a set with measure zero. So / / = 1 x m([0, l ] n ( R \ Q ) ) + 0 x m([0,1] n Q) = 1. Jo

LARGE AND SMALL

3.3

47

Large and Small

In this chapter we have looked at various ways to call a set large (and, of course, small). These notions are mostly unrelated to one another: A set could be small in one way (category) yet large in another (cardinality). We wish to end this chapter with a striking result that can be found in Oxtoby [58], taking the entire space we are working in and decomposing it into two small pieces. Theorem 3.20 The real line K can be partitioned into two disjoint sets A and B such that A is first category and B is measure zero. Proof: Start with r\, T2, r$,..., a listing of the rational numbers (it is left as a home­ work exercise to show Q is a countable set). For each Tj, let Uj,n be the interval

(

I T' —

i 3

1 2J+n '

V " ~~f~~ J

M

I

2-?+n) '

For each j let Oj = yj^L-JJ^n and then define B — f l ^ O j . For any e > 0 we can find an n so that 1/2" < e so B C UjUjtU and 2_j \Uj,n\ = Z^ 2^+"

=

2™


1. A simple continued fraction is an expression of the form 1

ooH

1

ai

a2 + a3 + '• To simplify our notation we will write this expression as [a0 :

ai,a2,a3,...}.

The individual values ao, a\, a2,. ■. are referred to as the partial quotients of the simple continued fraction expansion. S

EXAMPLE 4.1 The finite simple continued fraction [2:3,6] represents 2+

1

1 r = 2 + - r 7 r = 2H

3+i

f

6

19

44 = —.

19

This does, of course, work in the opposite direction, too. If we start with any rational number - where a, b £ Z, a ^ 0, then we can express it as a finite simple 1 Some sources allow complex numbers while other restrict the values to integers. Read these books and papers carefully.

INTRODUCING CONTINUED FRACTIONS

53

continued fraction. This requires just an application the Division Algorithm.2 For example, start with the number 32/7. We know 32 -f- 7 — 4 + 4/7 which is the same as Performing long division on 7/4 we have 1 + 3/4 so now our original number is 1 !+473

Next up, 4/3 = 1 + 1/3 and thus 32 7

1+

iil

[4:1,1,3].

77iere are two important features to point out. First, for a rational number in lowest terms this process is guaranteed to terminate. Notice that in our dividing process the remainders are strictly decreasing. This means they must at some point become the number 1. So although we are not writing this as a formal theorem, every rational number can be written as a finite simple continued fraction. Secondly, this is not the only way to represent 32/7. It is also correct that 32 - y

= 4 +

1 i

i

=[4:1,1,2,1].

However, the Division Algorithm, which guarantees that division gives unique re­ mainders, tells us that as long as we do not write the partial quotient an as an — 1 + \ we cannot have two representations. We shall then require in our representation [a0 : ai, a 2 , . . . , an] that an > 1. If our number ^ is negative, we change the first step so that we are multiplying the divisor by a negative number which will give us the smallest positive remainder. So, for instance, the continued fraction expansion of — 15/7 starts with —15 = —3 x 7 + 6. This returns to us -15 _ 1 ~ ~ ~ 3 + 7 / 6 ' and we continue to arrive at —15/7 = [—3 : 1,6] As we would expect, there are infinite simple continued fractions, where the pro­ cess continues indefinitely. It turns out that these very intriguingly partition into different types. An irrational number is one which cannot be written as p/q where p and q are integers, with q =/= 0. A special type is the quadratic irrational, a number of the form

p±Vd 2

Given two integers a and b with b j= 0, there exists unique integers q and r with 0 < r < \b\ such that a — bq + r.

54

CANTOR SETS AND CONTINUED FRACTIONS

where p, q, d are integers and d > 0 and not a perfect square. The name comes from the fact that it comprises the solutions to q2x2 - 2pqx + (p2 - d) = 0. The value TT, the all-purpose number, is an example of an irrational number that is not a quadratic irrational. Let x be a positive irrational number. Due to the Archimedean Property, there exists a largest integer ao such that x = a$ + — where 0 < (xi)^1 < 1. Then xx =

>1 x — ao and is irrational (after all, x is irrational and a$ is an integer). Repeat this process: Starting with x\, find a,\, the greatest integer such that 1

x\ = ai H

,

where 0 < ( a ^ ) - 1 < I. As we continue we generate the continued fraction for x [a0 : a i , a 2 , a 3 , . . . ] , where this time the sequence of'a, does not terminate as x is irrational. S

EXAMPLE 4.2 The easiest example is \/2. Given \/2 = 1.4142 . . . we see that ao = 1. Then \jx\ = \/2 — 1 or, taking reciprocals and rationalizing the denominator Xl='=-J—.y^±l

\/2-

1

V2-

1 A/2

+ 1

= V2 + l = 2.Ul2....

Thus ai = 2. The process then starts repeating, as X2 =

1 an-2

1

= —F=

A/2-1

Xi,

leading us to ai = 2 for i > 1. Hence V/2 = l +

^ 2

+

= [1:2,2,2,2,...].

1" 2+

2 + ■-

To shorten up the notation, since it is repeating, we follow the convention of using an overline, \/2 = [ 1 : 2 ] . ■

INTRODUCING CONTINUED FRACTIONS

55

This example shows why the quadratic irrationals are easy to deal with, as we can rationalize the denominator to simplify the process. Lagrange proved in 1779 that the continued fraction expansion of any quadratic irrational will eventually become periodic. It is also worth noting here a way in which continued fractions and rational numbers differ. If the decimal expansion of a number is periodic, then the number is rational. Having the continued fraction expansion repeat (as it does for \/2) does not guarantee a rational number. How about going the other direction ? When we have an ordinary real number but with a repeating decimal representation, we are able to find its rational number representation, can we do the same with a repeating continued fraction ? The answer is yes. Working backwards, we start with 1

[1:2] = 1

x7

where x is an unknown value. Using a trick similar to what we do with irrational numbers, then

x-i =

2+

.

K 2+ 2+ ••

We cannot just substitute into our equation willy-nilly. Using x — 1 to replace the entire second term in our definition of x leads to x = 1 + (x — 1) a tautology. This does not help us solve for x. Instead, we will substitute x — lfor the parenthetical part of 1

1

V 2+ 2+

• ■ . /

Thus we have the equations 1 2 + (a: - 1)

1 +

This equation simplifies to x2 = 2; and since we know we are dealing with the positive solution, it must be that x = \/2.

56

CANTOR SETS AND CONTINUED FRACTIONS

As another example, let us start with [1 : 1,1,1,...]. We set

i+

=i+ i

'

1+

x

I ^

1+ - 1 1 + '--

2

That yields x — x — 1 = 0 and the positive solution to this equation is

1 + V5

x =

, '

2

which is the golden ratio, usually denoted by . Thus c2i~i and " C2i_l > C2?;-2-

Thus the sequence of convergents has the property Ci < C3 < C5 < • • • < C2i-1 < • • • < C2i < • ■ • < C4 < C2-

This brings us, finally, to a theorem that says that any infinite simple continued fraction has meaning as a point on the real number line. Theorem 4.4 Let [ao : a,\, a2, 03,...] represent an infinite, simple continued frac­ tion. Then there is a point x on the real line such that x = [ao : aj, 112,03,...] Proof: Most of this proof has already been shown. We know that the two subse­ quences of convergents, {c2i} and {c2i-n}, are bounded and monotone. A wellknown theorem from real analysis (e.g., see [33]) is that monotone sequences con­ verge if and only if they are bounded. Thus {c2i} and {c2i+i} converge to limits Li and Li, respectively. Our job is to show that Li and L^ are the same. If not, then there exists an e > 0 such that Li — L2 > e, which means that for all i we have C2i ~ C2i+1 = l/(