Your quiz in recitation on Tuesday will cover 3.1: Arguments and inference. Your also have an online quiz, covering 3.1, due by 11:59 p.m., Tuesday.
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1 Friday, February 15 Today we will begin Course Notes 3.2: Methods of Proof. Your quiz in recitation on Tuesday will cover 3.1: Arguments and inference. Your also have an online quiz, covering 3.1, due by 11:59 p.m., Tuesday. Simple verbal arguments (from 3.1) Validity of symbolic arguments, using truth tables (from 3.1) More Rules of Inference, Fallacies (from 3.1) Inference, deduction (from 3.1) Constructing validity proofs (from 3.1) Rules of inference, quantifiers (from 3.1) Other quantified arguments (from 3.1) More complicated quantified arguments (from 3.1) Still more complicated quantified arguments (from 3.1) True or false: If n is any even number, then n 2 17 must be odd. A. True B. False
2 3.2 Methods of proof We now begin the practice of writing proofs in a context other than formal logic. In the definitions that follow, the assumption is that p and q are specific propositions about objects from mathematics and computer science. Direct proof To prove the proposition p q directly, we assume that the hypothesis p is true, and use that assumption to construct, step-by-step, a logical proof that q must be true. The steps of the proof use definitions, other established facts (theorems), and rules of inference. The proof is an incontrovertible explanation of why q must be true.
3 EXAMPLES 1. Use the definitions of even number, and odd number, to prove/disprove the following conjecture: If n is even, then n 2 17 is odd. First: Definitions An even integer or even number is an integer that is a multiple of 2; more formally: the integer n is even if and only if there exists an integer k such that n = 2k. An odd integer or odd number is the sum of an even number and 1; that is, the integer n is odd if and only if there exists an integer k such that n = 2k + 1. Our proof will proceed is follows: we will show, through calculation, that if n is any arbitrary integer that satisfies the definition of even number, then n 2 17 will satisfy the definition of odd number.
4 Closure Properties of the integers Our proof will also rely on the closure properties of the integers: The integers are closed under addition, subtraction, and multiplication. That is, the sum, difference and product of integers, is an integer. These closure properties are axioms (statements accepted as true, without proof) of the integers.
5 True or false: If x is any rational number, then 1/x is a rational number. A. True B. False
6 Prove or disprove the following conjecture: If x is a rational number and x 0, then 1/x is a rational number. First: Definition The real number x is a rational number if and only if there exist integers a, b, such that x = a/b in lowest terms, and b 0. Note: proving the conjecture will require a formal proof similar to the proof in the previous example. The formal proof will show that the conjecture is true for all nonzero rational numbers. On the other hand, if we suspect that the conjecture is false, then all that is needed to disprove it is a single counterexample. That is, to disprove this conjecture, we only have to find one nonzero rational number x for which the conjecture is not true.
7 Question: Are the rational numbers closed under addition or multiplication? A. The rational numbers closed under addition but not multiplication. B. The rational numbers closed under multiplication but not addition. C. The rational numbers closed under both addition and multiplication. D. The rational numbers closed under neither addition nor multiplication. For each of the two claims, if the claim is true, we should be able to write the proof using the definition of rational number. If the claim is false, we should be able to cite a counterexample proving that the claim is false.
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