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Axioms, Definitions, and Proofs

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Axioms, Definitions, and Proofs Proofs Definitions (An Alternative to Discrete Mathematics) G. Viglino Ramapo College of New Jersey January 2018 CONTENTS Chapter 1 A LOGICAL BEGINNING 1.1 Propositions 1 1.2 Quantifiers 14 1.3 Methods of Proof 23 1.4 Principle of Mathematical Induction 33 1.5 The Fundamental Theorem of Arithmetic 43 Chapter 2 A TOUCH OF SET THEORY 2.1 Basic Definitions and Examples 53 2.2 Functions 63 2.3 Infinite Counting 77 2.4 Equivalence Relations 88 2.5 What is a Set? 99 Chapter 3 A TOUCH OF ANALYSIS 3.1 The Real Number System 111 3.2 Sequences 123 3.3 Metric Space Structure of 135 3.4 Continuity 147 Chapter 4 A TOUCH OF TOPOLOGY 4.1 Metric Spaces 159 4.2 Topological Spaces 171 4.3 Continuous Functions and Homeomorphisms 185 4.4 Product and Quotient Spaces 196 Chapter 5 A TOUCH OF GROUP THEORY 5.1 Definition and Examples 207 5.2 Elementary Properties of Groups 220 5.3 Subgroups 228 5.4 Automorphisms and Isomorphisms 237 Appendix A: Solutions to CYU-boxes Appendix B: Answers to Selected Exercises PREFACE The underlying goal of this text is to help you develop an appreciation for the essential roles axioms and definitions play throughout mathematics. We offer a number of paths toward that goal: a Logic path (Chapter 1); a Set Theory path (Chapter 2); an Analysis path (Chapter 3); a Topology path (Chapter 4); and an Algebra path (Chapter 5). You can embark on any of the last three independent paths upon comple- tion of the first two: pter 3 Cha alysis h of An . A Touc y Ch 2 or A A To apter er he uch 4 pt T to of T a et u C opolo h S ch gy C of ha ch of p ou A te Chapter 1 T b r A st 5 .A Logical Beginning. ra c t A lg eb ra A one-semester course may include one, two, or even all three of the last three chapters, in total or in part, depending on the preparation of the students, and the ambition of the instructor. For our part, we have made every effort to assist you in the journey you are about to take. We did our very best to write a readable book, without compromising mathematical integrity. Along the way, you will encounter numerous Check Your Understanding boxes designed to challenge your understanding of each newly-introduced concept. Detailed solutions to each of the Check Your Understanding problems appear in Appendix A, but you should only turn to that appendix after making a valiant effort to solve the given prob- lem on your own, or with others. In the words of Desecrates: We never understand a thing so well, and make it our own, when we learn it from another, as when we have discovered it for ourselves. There are also plenty of exercises at the end of each section, covering the gamut of diffi- culty, providing you with ample opportunities to further hone your mathematical skills. Exercises that play a role in future developments are marked with a sharp (#). I wish to thank my colleague, Professor Marion Berger, for her numerous comments and suggestions throughout the development of the text. Her generous participation is deeply appreciated. I am also grateful to Professor Maxim Goldberg-Rugalev for his invaluable input. 1.1 Propositions 1 1 CHAPTER 1 A Logical Beginning Mathematics strives, as best it can, to transpose our physical universe into a non-physical form—a thought-universe, as it were, where defini- tions are the physical objects, and where the reasoning process rules. At some point, one has to ask whether we humans are creating mathemat- ics or whether, through some kind of wonderful mental telescope, we are capable of discovering a meager portion of the mathematical uni- verse which, in all of its glory, may very well be totally independent of us the spectators. Whichever; but one thing appears to be clear: the nature of mathematics, as we observe it or create it, appears to be inspired by what we perceive to be intuitive logic. §1. PROPOSITIONS We begin by admitting that “True” and “False” are undefined con- cepts, and that the word “sentence” is also undefined. Nonetheless: DEFINITION 1.1 A proposition (or statement) is a sentence PROPOSITION that is True or False, but not both. Fine, we have a definition, but its meaning rests on undefined words! We feel your frustration. Still, mathematics may very well be the closest thing to perfection we have. So, let’s swallow our pride and proceed. We can all agree that 53+8= is a True proposition, and that 53+9= is a False proposition. But how about this sentence: THE SUM OF ANY TWO ODD INTEGERS IS EVEN. First off, in order to understand the sentence one has to know all of its words, including: sum, integer, odd, and even. Only then can one hope to determine if the sentence is a proposition; and, if it is, to establish whether it is True or False. All in good time. For now, let us consider two other sentences: HE IS AN AMERICAN CITIZEN. This sentence cannot be evaluated to be True or False without first knowing exactly who “He” is. It is a variable proposition (more formally called a predicate), involving the variable “He.” Once He is specified, then we have a proposition which is either True or False. THE TWO OF US ARE RELATED. Even if we know precisely who the “two of us” are, we still may not have a proposition. Why not? Because the word “related” is just too vague. 2 Chapter 1 A Logical Beginning COMPOUND PROPOSITIONS The two numbers 3 and 5 can be put together in several ways to arrive at other numbers: 35+ , 35– , 35 , and so on. Similarly, two propo- sitions, p and q, can be put together to arrive at other propositions, called compound propositions. One way, is to “and-them:” In truth table form: DEFINITION 1.2 Let p, q be propositions. The conjunction of p q pq p and q p and q (or simply: p and q), written pq , T T T pq is that proposition which is True if both p and T F F q are True, and is False otherwise. F T F For example, the proposition: F F F 75 and 35+8= is True. (since both 75 and 35+8= are True) The proposition: 75 and 35+9= is False. (since 75 and 35+9= are not both True) Another way to put two propositions together is to “or-them:” In truth table form: DEFINITION 1.3 Let p, q be propositions. The disjunction of p p q pq and q (or simply: p or q), written pq , is T T T p or q T F T pq that proposition which is False when both p F T T and q are False, and is True otherwise. F F F For example, the proposition: 75 or 35+8= is True, as is the Note that p or q is True when both p and q are True. As such, it proposition 75 or 35+9= . The proposition 75 or 35+9= is said to be the inclusive-or. The exclusive-or is typically used in is False, since neither 75 nor 35+9= is True. conversation, as in: Do you want coffee or tea (one or the other, but not both) CHECK YOUR UNDERSTANDING 1.1 Let p be the proposition 75= , and q be the proposition 35+8= . (a) True (b) False (a) Is pq True or False? (b) Is pq True or False? The above and/or operators, which act on two given propositions, are said to be binary operators. The following operator is a unary operator in that it deals with only one given proposition: In truth table form: DEFINITION 1.4 Let p be a proposition. The negation of p, p p NOT P read “not p,” and written p , is that proposi- T F tion which is False if p is True, and True if p F T is False. For example, the proposition: ~7 5 is True, since the proposition 75 is False. The proposition: ~3+8 5= is False, since the proposition 35+8= is True. 1.1 Propositions 3 CHECK YOUR UNDERSTANDING 1.2 Indicate if the proposition p is True of False, given that (a) p is the proposition 53 . (a) False (b) False (b) p is the proposition q , where q is the proposition 35= . The negation operator “~” takes precedence (is performed prior) to both the “ ” and the “ ” operators. For example, the proposition: 35 35+9= is True. [since 35 is True] As is the case with algebraic expressions, the order of operations in a logical expression can be overridden by means of parentheses. Brackets”[ ]”play the same role as parentheses, and are For example, the proposition: used to bracket an expres- sion which itself contains 53 35+8= is False parentheses. [since 53 35+8= is True ] CHECK YOUR UNDERSTANDING 1.3 Assume that p and q are True propositions, and that s is a False prop- osition. Determine if the given compound proposition is True or False. (a) pq (b) pq (c) sq (d) sq True: d, f, g, h, j (e) pq (f) pq (g) sq (h) sq False: a, b, c, e, i, k (i) ps qs (j) ps qs (k) ps qs CONDITIONAL STATEMENT Consider the sentence: IF IT IS MONDAY, THEN JOHN WILL CALL HIS MOTHER.
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