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Introduction to Mathematical Reasoning Chris Woodward Rutgers

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Introduction to Mathematical Reasoning Chris Woodward Rutgers 7.1. Roster form and defining form 28 7.2. Subsets 29 7.3. Intersections and unions 29 Introduction to mathematical reasoning 7.4. Sets of subsets 31 7.5. Qualified quantifiers 31 7.6. Smullyan’s logic puzzles using sets 31 Chris Woodward 8. Proofs involving sets 32 Rutgers University, New Brunswick 8.1. Proving set containment and set equality 32 8.2. The set definition and roster axioms 33 Copyright 2013, All right reserved. 8.3. Problems with naive set theory 36 9. Proof strategies and styles 37 9.1. Working backwards 37 Contents 9.2. Proof summaries 39 1. Introduction 2 9.3. Writing proof summaries 39 2. Propositions and Connectives 2 9.4. Breaking proofs into pieces 41 2.1. Propositions 2 10. Ordered pairs and relations 43 2.2. Connectives and compound propositions 3 10.1. Ordered pairs and relations 43 2.3. Propositional Forms 4 10.2. Proofs involving relations 46 2.4. Conditionals and biconditionals 6 11. Functions 47 2.5. Smullyan’s logic puzzles 7 11.1. Functions 47 3. Predicates and Quantifiers 9 11.2. Injections, surjections, and bijections 48 3.1. Universal and existential quantifiers 9 11.3. Proofs involving functions 49 3.2. Working with quantifiers 10 11.4. Sequences and convergence to infinity 50 3.3. Quantifiers and numbers 11 11.5. Operations 51 3.4. Divisors and primes 12 12. Equivalence relations and partitions 53 3.5. Inequalities 13 12.1. Equivalence relations 53 4. Proofs by inference 14 12.2. Partitions and quotient sets 55 4.1. Inference and a few other rules 14 12.3. Cardinality 56 4.2. More rules of inference 16 13. Natural numbers and induction 58 5. Proof by deduction and contradiction 18 13.1. The induction axiom 58 5.1. Proof by deduction 18 13.2. The remainder theorem and base representation 60 5.2. Proof by contradiction 19 13.3. The Peano axioms 62 5.3. Arithmetic proofs by deduction or contradiction 21 14. Integers and primes 62 6. Proofs involving quantifiers 22 14.1. Properties of the integers 62 6.1. Universal instantiation and existential generalization 22 14.2. Prime numbers and prime factorizations 63 6.2. Universal generalization and existential instantiation 24 14.3. Common divisors 64 6.3. Proofs of inequalities 26 14.4. Fundamental theorem of arithmetic 65 6.4. More rules about quantifiers 27 14.5. Construction of the integers 67 7. Introduction to sets 28 14.6. Modular arithmetic 68 1 2 15. Rational numbers 69 sets, in an informal way in order to practice proofs; then as a set later. 15.1. Properties of the rational numbers 69 Induction does not appear until Section 13. 15.2. Construction of the rational numbers 70 Not all of the material is covered in a standard one-semester course. In 15.3. Suprema and infima 71 most years, I skip the material in the subsections on operations, groups, 15.4. Countability 71 the well-ordering principle and cardinality, although I talk briefly about 16. Limits of rational sequences 72 countability near the end of the course. 16.1. The definition of a limit 72 Finally, a warning about my conventions: 0 is a natural number, and 16.2. Properties of Limits 73 (f g)(x)= g(f(x)). 16.3. Base representations of rational numbers 74 ◦ 17. Real numbers 75 17.1. Problems with popular conceptions of real numbers 75 2. Propositions and Connectives 17.2. Real numbers as equivalence classes of bounded increasing sequences 76 2.1. Propositions. A statement that is either true or false will be called 17.3. Real numbers as equivalence classes of Cauchy sequences 76 a proposition. The truth value of a proposition is either true or false, 17.4. Existence of suprema and infima 77 depending on which it is. 17.5. Base representations of real numbers 77 Here are some examples of statements, propositions, and truth values. 17.6. Cantor’s uncountability argument 78 References 78 Statement Proposition? Truth Value 2+2=4 Yes True When ice melts, it turns into steam Yes False Chris Woodward played for the New York Mets Yes True 1. Introduction Alexander Hamilton was a U.S. President Yes False These notes cover what is often called an “introduction to proof” She is secretary of state No 2 course. In fact the main difficulty for students is translating their intu- x = 36 No itive ideas about what should be true into precise statements, especially This sentence is false No ones involving quantifiers. The beginning of the course, is inspired by an The fifth and sixth are not propositions because they are not suffi- approach that I learned from B. Kaufman, and described in the Elements ciently precise to have a definite truth value. That is, we don’t know of Mathematics series he co-authored [3]. whether x2 = 36 is true or false until we know what x is. The last state- Some of the idiosyncrasies of the notes are as follows. First, every ment is not a proposition because it cannot be either true or false. It rule of inference is given an abbreviation, but I don’t ask students to is called self-referential because it refers to itself. (Some self-referential memorize the abbreviations. Second, I couldn’t bring myself to com- statements can sometimes be considered propositions, but this isn’t one pletely identify P Q, Q can be derived from P , with the conditional of them.) statement P = ⊢Q; which are not the same by G¨odel. I note there is Whether a given sentence is a proposition can be, in some situations, a difference, but⇒ say that students may identify the two notions for the very debatable, depending on people agree on the meaning or not. For purposes of the course. Third, I mostly assume standard properties of example, I never make speling mistakes could be considered a proposi- the natural numbers, integers etc. The Peano axioms, and construction tion, if everyone agrees that “I” means the author of these notes. The of the integers and rationals as equivalence classes, are covered, but in a proposition, if it is one, is false. However, for the most part we will be non-crucial way. So for example, natural numbers are introduced before able to agree on which statements are propositions. 3 connectives such as and, or and not. For example, the author of these notes is the greatest teacher ever, or the Tampa Bay is going to win the world series, is a compound proposition. It might even be true! The statement The author of this book is an American and the Devil Rays are a baseball team is a compound proposition, which happens to be true, because of the two simple propositions that make up the compound the proposition both happen to be true. Some propositions are true or false because of the assertions they make about external reality, while others are true or false because of their inter- nal logical structure. The statement It is Tuesday and it is not Tuesday is a special kind of propositional, called a tautology. It is true because of its internal logical structure, and not because of external reality. The statement It is Tuesday and it is not Tuesday is a contradiction; it is false because of its internal logical structure. Problem 2.2. (From [1]) Which of the following are propositions, in your opinion? (Some are slightly debatable.) Comment on the truth values of all the propositions you encounter; if a sentence fails to be a Figure 1. He found the truth proposition, explain why. A statement can be a proposition regardless of whether we know its truth value. For an off-beat example, consider that the statement Colum- (1) All swans are white. bus was the first person to arrive in North America is definitely false, (2) The fat cat sat on the mat. Look in thy glass and tell whose face and off by about ten thousand years. Columbus was the first European thou viewest. My glass shall not persuade me I am old. to arrive in North America is also false, because of the Viking expedition (3) Father Nikolsky penned his dying confession to Patriarch Arsen to Newfoundland. But the statement Columbus was Jewish seems to be III Charnoyevich of Pe in the pitch dark, somewhere in Poland, unknown, even if we can agree what it means. Statements about the using a mixture of gunpowder and saliva, and a quick Cyrillic future or far past tend to have unknown truth-values. hand, while the innkeeper’s wife scolded and cursed him through Problem 2.1. Which of the following are propositions? (Some are de- the bolted door. batable.) Of those that are propositions, which have unknown (to you) (4) 1,000,000,000 is the largest integer. truth values? (5) There is no largest integer. There may or may not be a largest integer. (1) It rained on Columbus’ fifth birthday in Madrid. (6) Intelligent life abouds in the universe. (2) It will rain on Barack Obama’s sixtieth birthday in Hawaii. (7) This definitely is a proposition. (3) It is raining. (8) The speaker is lying. (4) If it is raining, then it is cloudy. (9) This is exercise number 12. 2.2. Connectives and compound propositions. A compound propo- (10) This sentence no verb. sition is a proposition formed from simpler propositions by the use of (11) ”potato” is spelled p-o-t-a-t-o-e. 4 Problem 2.3. Which of the following are propositions? (Some are (4) He can’t play violin or cello.
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