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Language and Proofs in Algebra: an Introduction Version 1.0

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Language and Proofs in Algebra: an Introduction Version 1.0 Language and Proofs in Algebra: An Introduction Version 1.0 c Faculty of Mathematics, University of Waterloo August 9, 2018 Contents 1 Introduction to the Language of Mathematics 6 1.1 The Language . 6 1.2 Sets........................................ 7 1.3 Mathematical Statements and Negation . 8 1.4 Quantifiers and Quantified Statements . 10 1.4.1 Universal and Existential Quantifiers . 10 1.4.2 Quantifiers and Language . 12 1.4.3 Negation of Quantifiers . 13 1.5 NestedQuantifiers................................ 14 1.5.1 Two Quantifiers . 14 1.5.2 An Arbitrary Number of Quantifiers . 15 1.5.3 Negation of Nested Quantifiers . 16 2 Logical Analysis of Mathematical Statements 19 2.1 Truth Tables and Negation . 19 2.2 Conjunction and Disjunction . 20 2.3 Logical Operators and Algebra . 22 2.4 Implication.................................... 25 2.5 Converse and Contrapositive . 29 2.6 IfandOnlyIf................................... 32 3 Proving Mathematical Statements 35 3.1 Proving Universally Quantified Statements . 36 3.2 ProvingExistentiallyQuantifiedStatements . 40 3.3 ProvingImplications............................... 43 3.4 DivisibilityofIntegers.............................. 47 3.4.1 Transitivity of Divisibility . 48 3.4.2 Divisibility of Integer Combinations . 50 3.5 Proof by Contrapositive . 54 3.6 Proof by Contradiction . 57 3.6.1ProvingUniqueness ............................ 59 3.7 ProvingIfandOnlyIfStatements . 60 4 Mathematical Induction 64 4.1 Notation for Summations, Products and Recurrences . 64 4.2 ProofbyInduction................................ 66 4.3 The Binomial Theorem . 72 4.4 Proof by Strong Induction . 77 2 Section 0.0 CONTENTS 3 5Sets 82 5.1 Introduction ................................... 82 5.2 Set-builder Notation . 83 5.3 SetOperations.................................. 86 5.4 SubsetsofaSet ................................. 87 5.5 Subsets, Set Equality and Implications . 89 6 The Greatest Common Divisor 93 6.1 The Division Algorithm . 93 6.2 The Greatest Common Divisor . 95 6.3 Certificate of Correctness and B´ezout’s Lemma . 98 6.4 The Extended Euclidean Algorithm . 101 6.5 Further Properties of the Greatest Common Divisor . 104 6.6 PrimeNumbers ................................. 108 6.7 The Unique Factorization Theorem . 109 6.8 Prime Factorizations and the Greatest Common Divisor . 112 7 Linear Diophantine Equations 116 7.1 The Existence of Solutions in Two Variables . 116 7.2 Finding All Solutions in Two Variables . 118 8 Congruence and Modular Arithmetic 122 8.1 Congruence . 122 8.2 Elementary Properties of Congruence . 123 8.3 Congruence and Remainders . 126 8.4 Linear Congruences . 131 8.5 Non-Linear Congruences . 133 8.6 Congruence Classes and Modular Arithmetic . 133 8.7 Fermat’s Little Theorem . 138 8.8 The Chinese Remainder Theorem . 141 8.9 SplittingaModulus ............................... 145 9 The RSA Public-Key Encryption Scheme 149 9.1 Public-Key Cryptography . 149 9.2 ImplementingtheRSAScheme. 150 9.3 Proving that the RSA Scheme Works . 154 10 Complex Numbers 157 10.1 Standard Form . 157 10.2 Conjugate and Modulus . 161 10.3 The Complex Plane and Polar Form . 165 10.4 De Moivre’s Theorem . 170 10.5 Complex n-thRoots............................... 173 10.6 Square Roots and the Quadratic Formula . 176 11 Polynomials 178 11.1 Introduction . 178 11.2 Arithmetic with Polynomials . 180 11.3 Roots of Complex Polynomials and the Fundamental Theorem of Algebra . 183 11.4 Real Polynomials and the Conjugate Roots Theorem . 188 11.5 Integer Polynomials and the Rational Roots Theorem . 190 4 Chapter 0 CONTENTS 11.6 More Examples for Roots and Factoring . 192 12 Additional Material 197 12.1 Prime Numbers and the Riemann Hypothesis . 197 Preface These course notes are meant to accompany the lectures of MATH 135 at the University of Waterloo. A number of faculty members in Mathematics have contributed to the writing and preparation over a number of years. This version is a revision for first use in Fall 2018. 5 Chapter 1 Introduction to the Language of Mathematics 1.1 The Language Mathematics is the language of mathematicians, and a proof is a method of com- municating a mathematical truth to another person who speaks the “language”. (Solow, How to Read and Do Proofs) Mathematics is an extraordinarily precise language. When we state a mathematical result, our aim is to leave no doubt about the meaning of that result, nor about what we could do in order to prove that it is true. Then, if we are able to prove that the result is true, our aim in writing a proof is to do so with no ambiguity, so there is once again no doubt about the correctness of our proof. The level of detail that is used depends on the audience, which we will assume consists of average students in the course. In this chapter we will concentrate on the language that is used in stating mathematical results, and on what would be required in order to determine that these results are true. To illustrate, we consider the following mathematical results. Example 1 For all integers n 5, ≥ 2n >n2. Example 2 There exists at least one integer a such that a2 + 29a + 209 0. The language used to state the two examples above is typical of the results that we will be considering in this course. What does this language mean? In Example 1, we are saying that the inequality 2n >n2 holds for each choice of the integer n whenever n 5. For example, when n = 5, we have 2n =25 = 32 and n2 =52 = 25, ≥ and 32 > 25, so the inequality holds in this case. But in order to prove that the first result 6 Section 1.2 Sets 7 is true, we would need to prove that the inequality holds for every possible choice of n. Verifying the inequality for values of n one at a time is not an e↵ective way to proceed. The methods of proof that we will learn in this course will allow us to learn how to handle situations of this type, and avoid simply verifying that some fact is true for every element of a given set. In Example 2, we are saying that the inequality a2 + 29a + 209 0 holds for at least one choice of the integer a. For example, when a = 0, we have a2 + 29a + 209 = 209, and clearly 209 > 0, so the inequality does not hold in this case. Trying again with a = 5, we have a2 + 29a + 209 = 379, and clearly 379 > 0, so the inequality does not hold in this case either. Trying once more with a = 5, we have a2 + 29a + 209 = 89, and 89 > 0, so again − the inequality does not hold in this case. How long should we try values of a at random? In order to prove that this result is true, we would only need to find one value of a for which the inequality holds. However, as for Example 1 above, checking the inequality for values of a one at a time is not in general an e↵ective way to proceed. The methods of proof that we will learn in this course will also allow us to precisely handle situations of this slightly di↵erent type. 1.2 Sets The mathematical results in Examples 1 and 2 involve the integers, which collectively form a set. Sets are fundamental in mathematics, and the way in which we refer to them forms an important part of the language of mathematics. Definition 1.2.1 A set is a collection of distinct objects. Each object that appears in this collection is called set, element an element (or member) of the set. Sets can contain any type of object. Since this is a mathematics course, we frequently use sets of numbers. But sets could contain letters, the letters of the alphabet for example, or books, such as those in a library collection. The simplest way to describe a set is to explicitly list all of its elements between a pair of brace brackets, and , separating the { } elements in the list by commas. Example 3 The following are examples of sets: 1. 2, 4, 6, 8 contains all the positive even numbers less that 10. { } 2. 1, 2, 1, 2, 3 is a set that contains three elements: 1, 2 and the set 1, 2, 3 . Note { { }} { } that the set 1, 2, 3 is considered a single element of 1, 2, 1, 2, 3 , even though { } { { }} 1, 2, 3 contains three elements itself. { } 3. , , , contains the symbols of the four suits in a deck of playing cards. {| ⌃ ~ } 4. The set of natural numbers, denoted by N, lists all the positive integers. That is, N = 1, 2, 3,... { } Note that this notation is not used consistently - in some parts of Mathematics and Computer Science the symbol N means the set of non-negative integers 0, 1, 2, 3,... { } 8 Chapter 1 Introduction to the Language of Mathematics 5. The set of integers, denoted by Z, lists all integers, whether they are negative, zero, or positive. That is, Z = ..., 2, 1, 0, 1, 2,... { − − } a 6. The set of rational numbers, denoted by Q, contains all numbers of the form b , where a is an integer and b is a non-zero integer. 7. The set of real numbers, denoted by R, contains all numbers in decimal form. Many of the mathematical results that we will encounter in this course concern the specially designated sets N, Z, Q and R described above. Except for these specially designated sets, usually we will use uppercase letters (S, T, U, etc.) to represent sets and lowercase letters (x, y, z, etc.) to represent elements of those sets. If x is an element of the set S,wewrite x S.Ifx is not an element of the set S,wewritex S. 2 62 Example 4 The following examples show how this notation is used: 1.
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