Discrete Mathematics I Lectures Chapter 4

1. Discrete Mathematics ILectures Chapter 4 Some material adapted from lecture notes provided by Dr. Chungsim Han and Dr. Sam Lomonaco Dr. Adam Anthony Spring 2011

2. Section 4.1 • Elementary Number Theory • Basic Proof Technique: Direct Proof

3. Tying It All Together: Self-Quiz • What is a valid argument? • When is a valid argument correct? When is it wrong? • How could you PROVE that a valid argument is correct?

4. What is a Proof? • A mathematical proof is a valid argument for which all premises are formally guaranteed (proven) to be true. • Different argument forms lead to different proof types • Sometimes guaranteeing a premise results in having to stop and prove the premise first!

5. Proof Terminology • An axiom is a basic assumption about mathematical structures that needs no proof. • We can use a proof to demonstrate that a particular statement is true. A proof consists of a sequence of statements that form an argument. • The steps that connect the statements in such a sequence are the rules of inference. • Cases of incorrect reasoning are called fallacies. • A theorem is a statement that can be shown to be true.

6. Proof Terminology • A lemma is a simple theorem used as an intermediate result in the proof of another theorem. • Lemmas prove premises to be true! • A corollary is a proposition that follows directly from a theorem that has been proved. • A conjecture is a statement whose truth value is unknown. Once it is proven, it becomes a theorem.

7. Elementary Number Theory • Some knowledge we’ll assume: • Properties of Real Numbers in Appendix A (‘basic algebra’) • All logic material from Chapters 2,3 • Properties of equality: • A = A (reflexive) • If A = B, then B = A (symmetric) • If A = B and B = C then A = C (transitive) • The Integers include …-2, -1, 0, 1, 2, … (No Fractions) • Integers are closed under +, -, * • Sum/Difference/Product of any two integers is also an integer • Integers are not generally closed under / • 3/2 is not an integer

8. Properties of Integers • An integer n is even  n = 2k for some integer k • An integer n is odd  n = 2k + 1 for some integer k • An integer n is prime   positive integers a and b, if n = ab, then a = 1 or b = 1 • An integer n is composite   positive integers a and b for which n = ab, and a 1 and b  1 • Using the symbol  means we can use these properties in either direction!

9. Exercises 1, 2 • If r is any integer (even or odd!), is (2r -1) odd? • If s is any integer, is 2s2 + 6s – even? • Which of the following are prime? 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 • Can an even number >2 be prime? • Is every integer > 1 either prime or composite?

10. Simplest Proof Type: Constructive Proof of Existence • You know this one already! • How do we prove: •  x in D, Q(x)? • Give a straight exampleProve that  even integer n such that n can be written in 2 ways as a sum of prime numbers • Give a set of instructions to find an example: Prove that  integer k, 22r + 18s = 2k

11. Simple ‘Proof’: Disproof by Counter Example • We know this one too! • How do we disprove • x in D, P(x)? • 2 identical approaches: • Directly find a single item in D for which P(x) is false • Negate the phrase and prove the negation (this one is a constructive proof!)

12. Disproof of a Conditional • A special case of disproof comes up again and again: • How do we disprove; • How do we disprove • x in D, P(x)  Q(x)? • In general, we can skip the above and just move to find an example where P(x) is true, but Q(x) is false.

13. Exercise 3 • Disprove the following statements: •  integers n, n2 – n + 11 is prime •  real numbers a and b, if a < b, then a2 < b2 • All primes are odd • For any integer n, if n2 = 4 then n = 2

14. Proving Universal Statements • We’ve discussed this before: disproving universal statements is easy, but proving them true is difficult! • In general, how do we prove • x in D, P(x)? • Exhaustive search: try everything in D! • What if D is infinite??? • Use a Generic Particular • Pick a variable x that has a particular but arbitrarily chosen value from the set D • Show that P(x) is true, without ever knowing what number was picked, based on the known axioms for items in D.

15. Disproving Existential Statements • How do we disprove: •  x in D, Q(x)? • 2 identical approaches: • Show that there isn’t a single item in D for which P(x) is false. • Negate the phrase and prove the negation using the previous slide: •  x in D, Q(x)  x in D, P(x)

16. Exercise 4 • True or False?There is an even integer n > 2 such that n is prime. • Perhaps the negation is easier to prove: • For any even integer n, if n > 2, then n is not prime • Proof: • Pick any even integer at all, we’ll call it x, and suppose that x > 2 • By definition of an even integer: x = 2m for some integer m • m must be greater than 1 since x > 2 • By definition of a prime integer, if x = a*b then either a or b must be equal to 1 • But x = 2m and 2  1 and m  1 • Therefore x is not prime (no matter what even integer you pick!) • If the negation is TRUE, then the original statement must be FALSE!!!

17. Proofs Using Universal Modus Ponens • x in D, P(x)  Q(x)P(c) for a particular c in DQ(c) • Proof: MUST SHOW THAT ALL PREMISES ARE GUARANTEED TO BE TRUE! • Then, the conclusion follows as a proof • Proving P(c): easy—it’s either true or false for that one thing • How do we prove x in D, P(x)  Q(x) is true?

18. Direct Proof • Many proofs start with or otherwise use a universally quantified conditional statement: • x in D, P(x)  Q(x) • One way to show this is true is using a Direct Proof: • Suppose P(x) is true for some particular but arbitrarily chosen element in D. • Show that Q(x) must be true by using definitions, previously established results, and rules for logical inference.

19. Tips on Getting Started with a Proof that P(x)  Q(x) • Read and re-read the question until you clearly understand what is being asked: • Write down the starting point, adapted from the left-hand side of the implication—“Suppose there exists a particular but arbitrarily chosen m for which P(m) is true.” • Write down a clear statement for what you need to prove: “Therefore, Q(m) is true” • On the side, write down every definition and axiom that you know which you think might be relevant • From the starting point, use the definitions and axioms to build up toward the conclusion

20. Exercise 5 • Prove that, for all Integers n, if n is even, then n2is even. • Starting Point: Suppose n is a [particular but arbitrarily chosen] integer such that n is even. • By definition, n = 2k for some integer k • Then n2 = (2k)2 = 4k2 = 2(2k2) using basic algebra. • Then n2 = 2m where m = 2k2, and 2k2 is an integer. • [Conclusion to be shown: ] Therefore, by definition n2is even

21. Exercise 6 • Prove that for all integers m and n, if m is even and n is odd, then m + n is odd

22. General Guidelines for Proof Writing • Copy the statement of the theorem to be proved onto the paper • Clearly mark the beginning of the proof with the word ‘Proof’. • Make your proof self-Contained • Explain the meaning of each variable you use • Write your proof in complete, grammatically correct sentences, mathematical notation excepted. • “Then M + N = 2r + 2s.”

23. General Guidelines for Proof Writing • Be clear about assumptions or things that haven’t been proved yet. • Give a reason for each statement • By definition, by hypothesis, by previous theorem, etc. • Use good connecting words to guide logic • Then, thus, so, hence, It follows that, therefore, etc. • DISPLAY equations and inequalities • Put them on a SEPARATE line, centered. • White space is your friend, improves readability • PROOFS IN YOUR JOURNAL DO NOT HAVE TO FOLLOW THESE GUIDELINES! • Journal work may be as messy as you like! • You don’t have to use complete sentences, give reasons, or even be clear about assumptions and such • It’s the copying phase—where you re-write for the homework where you apply these 8 guidelines!

24. Common Proof Mistake: Arguing from examples • “Proof” that  integers n, n2 – n + 11 is prime • If n = 6, then 62 – 6 + 11 = 41 which is prime so this must work • Using examples is useful when you are thinking through the problem (esp. in your journal) to visualize structure • But in general, proofs must work for ALL items in the domain, so picking 1 or 2 is not satisfactory proof.

25. Common Proof Mistake: Using the same letter for two different values • “Suppose m and n are any odd integers. Then by definition of odd, m = 2k + 1 and n = 2k + 1 for some integer k.” • Why is this wrong?

26. Common Proof Mistake: Jumping to a Conclusion and Circular Reasoning • “Proof” that the sum of two even integers is even: • Suppose m and n are any even integers. By definition of even, m = 2r and n = 2s for some integers r and s. Then m + n = 2r + 2s. So m + n is even. • Jumping to a conclusion: the last line does not fit any definition! • “Proof” that the product of two odd integers is odd: • Suppose m and n are any odd integers. When any odd integers are multiplied, their product is odd. Hence mn is odd. • See the circular reasoning here?

27. Section 4.3: Divisibility • Definition: If n and d are integers and d  0 then n is divisible by d if, and only if, n equals d times some integer • Identical phrases: n is a multiple of d, d is a factor of n, d is a divisor of n, d divides n • Notation: • d | n is read as ‘d divides n’ and means: •  integer k, n = dk • OR, n/k is an integer • d ∤ n is read as ‘d does not divide n’ and means: • integer k, n  dk • OR, n/k is not an integer

28. Exercise 1 • Is 18 divisible by 6? • Does 3 divide 12? • Does 5 | 25? • If d is a nonzero integer, does d divide 0? • If m and n are integers is 10m + 15n divisible by 5? • Does 6 | 3?

29. Exercise 2 • Prove that for all integers a, b and c, ifa|bandb|cthena|c. [This is the transitive property of divisibility]

30. Exercise 3 • Prove that for all integers a and b if a|b then a2|b2.

31. Exercise 4 • Disprove the following: For all integers a and b if a|b and b|a then a = b.

32. Section 4.4: The Quotient-Remainder Theorem • Think of an integer as a set of whole items: • 12 = xxxxxxxxxxxx • Then saying 12  3 is the same as dividing the set into groups of size 3: • 12  3 = xxx xxxxxxxxx, • 4 groups of 3 • 12 = 4*3 • What about 13  3? • 13  3 = xxx xxxxxxxxx x, • 4 groups of 3, with one left over • 13 = 4*3 + 1

33. The Quotient Remainder Theorem • For any integer n and a positive integer d, there exist unique integers q and r such that: n = dq + r and 0 r<d • Find values for q and r for the given n and d: • n = 26, d = 7 • n = -26 and d = 7 • n = 27 and d = 30 • n = -34 and d = 11

34. Definitions, notation • n div d: the value of q (aka the quotient)when n is divided by d • n mod d: the value of r (aka the remainder)when n is divided by d • Find: • 29 div 8 • 29 mod 8 • -29 div 8 • -29 mod 8

35. Applying The Quotient-Remainder Theorem • Apply the QR theorem to any integer n with d = 2 • Just fill in the blanks with what we know, leaving the rest: • For any integer n and the positive integer 2, there exist unique integers q and r such that: • n = 2q + r and 0 r<2 • What can we conclude??? • The parity property refers to the fact that any given integer is either even or odd, and not both.

36. Dealing With Uncertainty • Up to now, if we didn’t know something for sure then we ignored it • But we don’t have to! • Take any integer n: • Is it even or odd? • If it is even, then we know certain facts about n • If it is odd, then we know a different set of facts These two ‘possible scenarios’ are referred to as cases

37. Division into Cases—Argument Form p  r q  r s  r … z  r (p  q  s  …  z) r Cases where r must be true Assertion that at least one of the cases MUST have occurred.

38. Division into Cases—Proof Form • Given an item r to prove in which some item can fall into several categories (e.g., odd/even): • Assert that there exist a fixed number of potential cases in the world that can be true: (Case1  Case2 …) • Set up the following arguments: • Case1  r • Case2  r • … • Perform a direct proof for each case above • If you succeed for all cases, then by valid argument, r must always be true.

39. Proof Example 1 • Prove that the product of any two consecutive integers is even. • Starting Point: Select any integer n. • Need to prove: the product of n and the next consecutive integer (n + 1) is even. • What do we know for sure about n? • What don’t we know for sure?

40. Proof Example 1 • Prove that the product of any two consecutive integers is even. • Select any integer n. • Case 1: n is even. • Case 2: n is odd.

41. Proof Example 2 • Prove that if n is an integer, then n2 n • Common set of cases: n = 0, n < 0, n > 0 • ‘Divide and conquer’ approach: break a big problem into 2 or more smaller problems.

42. Proof Example 3 • Show that |xy| = |x||y| where x and y are real numbers. • With two numbers, what are all the possible scenarios (cases)? Absolute Value: |x| = x when x  0 |x| = -x when x < 0

43. Proof Example 4: Triangle Inequality • Show that for all real numbers x and y, |x + y|  |x| + |y| Absolute Value: |x| = x when x  0 |x| = -x when x < 0

44. Section 4.6: Indirect Proof Techniques • Proof by contraposition • Proof by contradiction

45. First attempt • Let’s try to use a direct proof for the following: • Prove for all integers n, if n2 is even, then n is even

46. The Contrapositive Trick • Remember that an implication’s contrapositive is equivalent to the original statement: • p  q  q  p • So for any proof statement, we can replace an implication with its contrapositive and prove that instead! • The proof of the new statement is a necessary and sufficient proof of the original

47. Second Attempt • Use contraposition this time: • Prove for all integers n, if n2 is even, then n is even • Contrapositive version: for all integers n, if n is not even, then n2 is not even • If n is odd, then n2 is odd Prove this!

48. Contraposition Example 1 • Prove that for all integers n, if 3n+2 is odd, then n is odd.

49. Contraposition Example 2 • For all integers n, if 5 ∤ n2 then 5 ∤ n

50. Proof by Contradiction • An inference rule: • p (p  c) • What is the intuition here? • To use this to our advantage, we use the ‘cause-effect’ interpretation of : • If p were true, then it MUST follow that some contradiction arises • If we can show that, then (p  c) is true, which means p is true