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Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis

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Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis Mike Krebs James Pommersheim Anthony Shaheen FOR THE INSTRUCTOR TO READ i For the instructor to read We should put a section in the front of the book that organizes the organization of the book. It would be the instructor section that would have: { flow chart that shows which sections are prereqs for what sections. We can start making this now so we don't have to remember the flow later. { main organization and objects in each chapter { What a Cfu is and how to use it { Why we have the proofcomment formatting and what it is. { Applications sections and what they are { Other things that need to be pointed out. IDEA: Seperate each of the above into subsections that are labeled for ease of reading but not shown in the table of contents in the front of the book. ||||||||| main organization examples: ||||||| || In a course such as this, the student comes in contact with many abstract concepts, such as that of a set, a function, and an equivalence class of an equivalence relation. What is the best way to learn this material. We have come up with several rules that we want to follow in this book. Themes of the book: 1. The book has a few central mathematical objects that are used throughout the book. 2. Each central mathematical object from theme #1 must be a fundamental object in mathematics that appears in many areas of mathematics and its applications. 3. When definitions of abstract objects such as functions and equivalence relations come up the book focuses on the central examples. 4. Add more... 5. Add more... We chose two central themes for the book: number theory / abstract algebra, and analysis / topology. For number theory and abstract algebra, the main examples are the integers modulo n, properties of the prime numbers, etc. Some appli- cations sections include sections on generating the pythagorean triples, the group of primitive pythagorean triples, the Guassian integers with an application to sums of squares. For analysis / topology, we study the real line and real plane. Fill this part in .... distance function? open sets/closed sets? See the flow chart in Figure ... The chapter contents are layed out as follows. FOR THE INSTRUCTOR TO READ ii Ch. 1 - Ch. 2 - Ch. 3 - This is really where the nuts and bolts of the course start. We go through the kinds of proofs that one encounters in math texts, such as direct proof, contradiction, etc. We discuss the divisibility properties of the integers and focus on congruence modulo n. Prime numbers, irrational numbers, and the greatest common divisor are are introduced in this chapter and are used throughout the text. Powerful theorems about prime numbers, such as Theorem and Theorem on page and page , are provenp in this chapter. Their first application appears in the proof that 3 is irrational. Ch. 4 - We introduce the methods of induction. We prove two big theorems from number theory: There are an infinite number of primes, and every integer can be factored uniquely into a product of primes. Ch. 5 - sets Ch. 6 - relations. We focus on equivalence relations and equivalence classes because it is the most important topic in the chapter. One main example that is used throughout the remainder of the book is the integers modulo n. We especially use the integers modulo n in the functions chapter. Ch. 7 - functions. Functions are one of the most important ob- jects in mathematics. In illustrating concepts such as one-to-one, onto, image, and inverse image we use examples from our central objects: in- tegers modulo n, integers, whatever else we add in this chapter. There are several examples of functions between the integers modulo n that are used in group theory and number theory. Put what the real number ones will be if we add more or replace some of the number theory. ... ... ... ... Here are several ways to navigate through the book: Way 1: Put a sequence of sections here. Way 2: Put a sequence of sections here. Contents For the instructor to read i Chapter 1. Introduction 1 1.1. Welcome to mathematics! 1 1.2. What are proofs, and why do we do them? 9 Part 1. Basics 14 Chapter 2. Basics of sets 15 2.1. Beginning to work with sets 15 2.2. Exercises 23 Chapter 3. Logic 26 3.1. Rules of logic 26 3.2. Quantifiers 33 3.3. Exercises 36 3.4. Fun math facts 38 Part 2. Main Stuff 39 Chapter 4. Proof techniques 40 4.1. Our starting assumptions 40 4.2. Proofs from Axioms { put as optional at end 45 4.3. Direct Proofs 50 4.4. Proofs by cases 54 4.5. Existence and Uniqueness Proofs 55 4.6. Contraposition 56 iii CONTENTS iv 4.7. Contradiction 58 4.8. If and only if proofs 60 4.9. Proofs involving nested quantifiers 61 4.10. Application: Number Theory 61 4.11. Exercises 65 4.12. Fun math facts 68 Chapter 5. Induction 70 5.1. Proofs by induction 70 5.2. Complete induction 72 5.3. Applications to number theory 74 5.4. Exercises 76 5.5. Fun math facts 77 Chapter 6. Sets 79 6.1. Subsets 79 6.2. Unions, intersections, and complements 86 6.3. Cartesian products 93 6.4. Power sets, and sets as elements 101 6.5. Families of sets 103 6.6. Exercises 110 6.7. Fun math facts 112 Chapter 7. Relations 113 7.1. Relations 113 7.2. Equivalence Relations 114 7.3. Integers modulo n 118 7.4. Well-defined operations 122 7.5. Partitions 126 7.6. Partial Orders 126 7.7. Exercises 126 Chapter 8. Functions 130 8.1. Functions 130 8.2. Well-defined functions 138 8.3. One-to-one and Onto functions 140 CONTENTS v 8.4. Composition of functions 151 8.5. Inverse functions 156 8.6. Image and pre-image of a function 161 8.7. Application: Pythagorean triples 162 8.8. Exercises 173 Chapter 9. Cardinality 186 9.1. Cardinality of a set 186 9.2. Finite sets 188 9.3. Countable sets 189 9.4. Uncountable sets 192 9.5. Cantor-Schroeder-Bernstein Theorem 192 9.6. Exercises 192 9.7. Fun math facts. 192 Part 3. Extras 193 Chapter 10. Number Theory 194 10.1. Gaussian integers 194 10.2. Division and primes 196 10.3. Integers modulo a prime 199 10.4. Sums of squares 202 10.5. Exercises 204 Chapter 11. Real Analysis 205 11.1. Supremum of a set 205 11.2. Limits 209 11.3. Infinite Sums 215 11.4. Bounded monotone convergence theorem and the irrationality of e 216 11.5. Exercises 219 Chapter 12. Group Theory 221 12.1. Definition of a group 221 12.2. The symmetric group 224 12.3. The group of Pythagorean triples 224 12.4. Exercises 231 CONTENTS vi Chapter 13. The Standard Number Systems 232 13.1. The natural numbers 233 13.2. The integers 251 13.3. The rationals 265 13.4. The reals 272 13.5. The complex numbers 293 13.6. Properties of the number systems 298 13.7. Cauchy sequences 299 13.8. Fun math facts 304 Chapter 1 Introduction 1.1. Welcome to mathematics! We expect that most people reading this book will be students taking a transitional \proofs" course; that most such students are math majors; and that this is the first course exclusively for, or at least dominated by, math majors. So we feel that this is a great opportunity to welcome and introduce you to the wide and wonderful world of mathematics. As you have probably heard, lower-division math courses such as Calculus tend to be heavily computational, whereas upper-division math courses are usually more theoretical and abstract. The primary purpose of this text is to help you make that transition. In this first section, though, we'd like you to tell you about some fun and useful pieces of information that together comprise a significant portion of \math culture." Your professors probably know most of this, plus lots more that we overlooked, but as an early-career math major, chances are that you have not yet been clued in. Though many of the items below apply throughout the globe, some pertain only to the United States, where the authors of this book reside. Moreover, the longer it's been since this section was written, the more likely it is to be obsolete. An internet search should provide information that is up-to-date and relevant to countries outside the US. But now you'll know some things you can look for! Jobs in math: Math majors are often not aware of job opportunities outside of teaching that make use of their academic training. Such ca- reer paths abound, however. Moreover, they are not just jobs; they are 1 1. INTRODUCTION 2 highly desirable jobs. In the year 2014, CareerCast.com rated \math- ematician" as the very best in a ranking of two hundred different jobs. Criteria included work environment, income, stress level, etc. Many other highly rated jobs were math-related, such as actuary and statis- tician, both of which were in the top five. Math-intensive vocations include, but are by no means limited to, the following: actuary, cryptologist, economist, environmental mathe- matician, geodesist, inventory strategist, operations research analyst, staff systems air traffic control analyst, and statistician. For more information, you may want to visit the Mathematical Association of America's Career Profiles website at: www.maa.org/careers/career-profiles Teaching math: Teaching, of course, remains a way to make use of a degree in mathematics.
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