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Mathematical Reasoning & Proofs

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Mathematical Reasoning & Proofs Mathematical Reasoning & Proofs MAT 1362 Fall 2021 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Contents Preface 4 1 Integers 5 1.1 Axioms.......................................5 1.2 First consequences of the axioms........................6 1.3 Subtraction.................................... 13 2 Natural numbers and induction 15 2.1 Natural numbers................................. 15 2.2 Ordering the integers............................... 16 2.3 Induction..................................... 20 2.4 The well-ordering principle............................ 24 3 Logic 27 3.1 Quantifiers..................................... 27 3.2 Implications.................................... 29 3.3 Negations..................................... 31 4 Finite series and strong induction 34 4.1 Preliminaries................................... 34 4.2 Finite series.................................... 36 4.3 The Binomial Theorem............................. 40 4.4 Strong induction................................. 43 5 Naive set theory 46 5.1 Subsets and equality............................... 46 5.2 Intersections and unions............................. 49 5.3 Cartesian products................................ 53 5.4 Functions..................................... 54 5.5 Russell's paradox and axiomatic set theory................... 55 6 Equivalence relations and modular arithmetic 57 6.1 Equivalence relations............................... 57 6.2 The division algorithm.............................. 61 6.3 The integers modulo n .............................. 63 6.4 Prime numbers.................................. 68 2 Contents 3 7 Real numbers 73 7.1 Axioms....................................... 73 7.2 Positive real numbers and ordering....................... 76 7.3 The real numbers versus the integers...................... 79 7.4 Upper and lower bounds............................. 81 8 Injections, surjections, and bijections 87 8.1 Injections, surjections, and bijections...................... 87 8.2 Embedding Z in R ................................ 94 9 Limits 95 9.1 Unboundedness of the integers.......................... 95 9.2 Absolute value.................................. 96 9.3 Distance...................................... 99 9.4 Limits....................................... 100 9.5 Square roots.................................... 108 10 Rational and irrational numbers 109 10.1 Rational numbers................................. 109 10.2 Irrational numbers................................ 111 Index 113 Preface These are notes for the course Mathematical Reasoning & Proofs at the University of Ot- tawa. This is an introduction to rigorous mathematical reasoning and the concept of proof in mathematics. It is a course designed to prepare students for upper-level proof-based mathematics courses. We will discuss the axiomatic method and various methods of proof (proof by contradiction, mathematical induction, etc.). We do this in the context of some fundamental notions in mathematics that are crucial for upper-level mathematics courses. These include the basics of naive set theory, functions, and relations. We will also discuss, in a precise manner, the integers, integers modulo n, rational numbers, and real numbers. In addition, we will discuss concepts related to the real line, such as completeness, supremum, and the precise definition of a limit. In general, this course is designed to provide students with a solid theoretical foundation upon which to build in subsequent courses. This course should be of interest to students who feel dissatisfied with mathematics courses in which computation is emphasized over proof and explanation. In MAT 1362, we will rarely perform computations based on standard techniques and algorithms, such as computing derivatives using the chain and product rules. Instead, we will focus on why certain mathematical statements are true and why given techniques work. This is often much more difficult than following an algorithm to perform a computation. However, itis much more rewarding and results in a deeper understanding of the material. Acknowledgement: Significant portions of these notes closely follow the book The Art of Proof by Matthias Beck and Ross Geoghegan [BG10], which is the official course text. Alistair Savage Course website: https://alistairsavage.ca/mat1362 4 Chapter 1 Integers In this chapter we discuss the integers. While you have encountered the integers in many previous courses, dating back to grade school, you probably took certain properties for granted. In this course, we will begin with some precise axioms and deduce other properties of the integers in a more rigorous way. 1.1 Axioms We will discuss the concept of a set later, in Chapter5. For now, we will use the word intuitively. A set S is a collections of things, called the elements or members of S. We write a 2 S to indicate that a is an element of S.A binary operation on a set S is a function that takes two elements of S as input and produces another element of S as output. We will assume there is a set Z, whose members we will call integers. This set comes with two binary operations: • addition, denoted +, and • multiplication, denoted ·. We will often denote multiplication by juxtaposition. That is, we will write ab instead of a · b. We assume that Z, together with these operations, satisfies the following five axioms, as well as Axioms 2.1 and 2.17 to be introduced in Chapter2. Axiom 1.1 (Commutativity, associativity, and distributivity). For all integers a, b, and c, we have (i) a + b = b + a,(commutativity of addition) (ii)( a + b) + c = a + (b + c), (associativity of addition) (iii) a · (b + c) = a · b + a · c,(distributivity) (iv) a · b = b · a,(commutativity of multiplication) (v)( a · b) · c = a · (b · c). (associativity of multiplication) 5 6 Integers Axiom 1.2 (Additive identity). There exists an integer 0 such that a + 0 = a for all a 2 Z. This element 0 is called an additive identity, or an identity element for addition. Axiom 1.3 (Multiplicative identity). There exists an integer 1 such that 1 6= 0 and a · 1 = a for all a 2 Z. The element 1 is called a multiplicative identity, or an identity element for multiplication. Axiom 1.4 (Additive inverse). For each a 2 Z, there exists an integer, denoted −a, such that a + (−a) = 0. The element −a is called the additive inverse of a. Axiom 1.5 (Cancellation). If a; b; c 2 Z, a · b = a · c, and a 6= 0, then b = c. This is called the cancellation property. The symbol \=" means equals. When we write a = b for elements a and b of some set S (e.g. Z), we mean that a and b are the same element. This symbol has the following properties: (i) a = a.(reflexivity of equality) (ii) If a = b, then b = a.(symmetry of equality) (iii) If a = b and b = c, then a = c.(transitivity of equality) (iv) If a = b, then a can be replaced by b in any statement or expression, without changing the meaning of that statement or expression. (In fact, the statement/expression is unchanged.) We call this the replacement property. For example, if a; b; c 2 Z and a = b, then a + c = b + c. The symbol 6= means is not equal to. When we write a 6= b for a; b 2 Z, we mean that a and b are not the same element of Z. Note the following: (i) The symbol 6= is not reflexive. We never have a 6= a. (ii) The symbol 6= is symmetric. If a; b 2 Z and a 6= b, then b 6= a. (iii) The symbol 6= is not transitive. For example, 1 6= 2 and 2 6= 1, but it is not true that 1 6= 1 (which is what transitivity would imply). The symbol 62 means is not an element of. 1.2 First consequences of the axioms For now, we only assume Axioms 1.1{1.5. That is, we will not use any other properties of the integers that you may have learned in previous courses in mathematics. We will deduce other properties of the integers from these axioms. Once we have proved that some other statement is true, we may then use it in following statements. In general, we would like to assume as few axioms as possible, and show that other properties are implied by this small list of axioms. Note, for example, that we have not yet introduced the operation of subtraction. (We will see this in Section 1.3.) First consequences of the axioms 7 Proposition 1.6. If a; b; c 2 Z, then (a + b)c = ac + bc. Proof. Suppose a; b; c 2 Z. Then we have (a + b)c = c(a + b) Axiom 1.1(iv) (commutativity of multiplication) = ca + cb Axiom 1.1(iii) (distributivity) = ac + bc: Axiom 1.1(iv) (commutativity of multiplication) Proposition 1.7. If a 2 Z, then 0 + a = a and 1 · a = a. Proof. This proof of this proposition is left as an exercise (Exercise 1.2.1). Proposition 1.8. If a 2 Z, then (−a) + a = 0. Proof. The proof of this proposition is left as an exercise (Exercise 1.2.2). Proposition 1.9. If a; b; c 2 Z and a + b = a + c, then b = c. Proof. Suppose a; b; c 2 Z. Then we have a + b = a + c =) (−a) + (a + b) = (−a) + (a + c) replacement property =) ((−a) + a) + b = ((−a) + a) + c Axiom 1.1(ii) (associativity of addition) =) 0 + b = 0 + c Proposition 1.8 =) b = c: Proposition 1.7 Above, we have used the symbol =) . If P and Q are two statements, then P =) Q means that the statement P implies the statement Q or, equivalently, that Q follows logically from P . It can also be read \if P , then Q". Proposition 1.10 (Uniqueness of the additive inverse). If a; b 2 Z and a + b = 0, then b = −a. In other words, the element −a mentioned in Axiom 1.4 is the unique additive inverse of a (i.e. every integer has exactly one additive inverse). Proof. Suppose a; b 2 Z. Then we have a + b = 0 =) (−a) + (a + b) = (−a) + 0 replacement property =) (−a) + (a + b) = −a Axiom 1.2 (add.
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