CS322

1. Week 3 - Friday CS322

2. Last time • What did we talk about last time? • Proving universal statements • Disproving existential statements • Rational numbers • Proof formatting

3. Questions?

4. Logical warmup • An anthropologist studying on the Island of Knights and Knaves is told that an astrologer and a sorcerer are waiting in a tower • When he goes up into the tower, he sees two men in conical hats • One hat is blue and the other is green • The anthropologist cannot determine which man is which by sight, but he needs to find the sorcerer • He asks, "Is the sorcerer a Knight?" • The man in the blue hat answers, and the anthropologist is able to deduce which one is which • Which one is the sorcerer?

5. Logical cooldown • After determining that the man in the green hat is the sorcerer, the sorcerer asks a question that has a definite yes or no answer • Nevertheless, the anthropologist, a naturally honest man, could not answer the question, even though he knew the answer • What's the question?

6. Rational Numbers Review

7. Another important definition • A real number is rational if and only if it can be expressed at the quotient of two integers with a nonzero denominator • Or, more formally, r is rational   a, b Z  r = a/b and b  0

8. Prove or disprove: • The reciprocal of any rational number is a rational number

9. Using existing theorems • Math moves forward not by people proving things purely from definitions but also by using existing theorems • "Standing on the shoulders of giants" • Given the following: • The sum, product, and difference of any two even integers is even • The sum and difference of any two odd integers are even • The product of any two odd integers is odd • The product of any even integer and any odd integer is even • The sum of any odd integer and any even integer is odd • Using these theorems, prove that, if a is any even integer and b is any odd integer, then (a2 + b2 + 1)/2 is an integer

10. Divisibility

11. Definition of divisibility • If n and d are integers, then n is divisible by d if and only if n = dk for some integer k • Or, more formally: • For n, d  Z, • n is divisible by d  k Z  n = dk • We also say: • n is a multiple of d • d is a factor of n • d is a divisor of n • d divides n • We use the notation d | n to mean "d divides n"

12. Transitivity of divisibility • Prove that for all integers a, b, and c, if a | b and b | c, then a | c • Steps: • Rewrite the theorem in formal notation • Write Proof: • State your premises • Justify every line you infer from the premises • Write QED after you have demonstrated the conclusion

13. Prove or disprove: • For all integers a and b, if a | b and b | a, then a = b • How could we change this statement so that it is true? • Then, how could we prove it?

14. Unique factorization theorem • For any integer n > 1, there exist a positive integer k, distinct prime numbers p1, p2, …, pk, and positive integers e1, e2, …, ek such that • And any other expression of n as a product of prime numbers is identical to this except, perhaps, for the order in which the factors are written

15. An application of the unique factorization theorem • Let m be an integer such that • 8∙7 ∙6 ∙5 ∙4 ∙3 ∙2 ∙m = 17∙16 ∙15 ∙14 ∙13 ∙12 ∙11 ∙10 • Does 17 | m? • Leave aside for the moment that we could actually compute m

16. Quotient Remainder Theorem and Proof by Cases

17. Proof by cases • If you have a premise consisting of clauses that are ANDed together, you can split them up • Each clause can be used in your proof • What if clauses are ORed together? • You don't know for sure that they're all true • In this situation, you use a proof by cases • Assume each of the individual possibilities is true separately • If the proof works out in all possible cases, it still holds

18. Proof by cases formatting • For a direct proof using cases, follow the same format that you normally would • When you reach your cases, number them clearly • Show that you have proved the conclusion for each case • Finally, after your cases, state that, since you have shown the conclusion is true for all possible cases, the conclusion must be true in general

19. Quotient-remainder theorem • For any integer n and any positive integer d, there exist unique integers q and r such that • n = dq + r and 0 ≤ r < d • This is a fancy way of saying that you can divide an integer by another integer and get a unique quotient and remainder • We will use div to mean integer division (exactly like / in Java ) • We will use mod to mean integer mod (exactly like % in Java) • What are q and r when n = 54 and d = 4?

20. Even and odd • As another way of looking at our earlier definition of even and odd, we can apply the quotient-remainder theorem with the divisor 2 • Thus, for any integer n • n = 2q + r and 0 ≤ r < 2 • But, the only possible values of r are 0 and 1 • So, for any integer n, exactly one of the following cases must hold: • n = 2q + 0 • n = 2q + 1 • We call even or oddness parity

21. Consecutive integers have opposite parity • Prove that, given any two consecutive integers, one is even and the other is odd • Hint Divide into two cases: • The smaller of the two integers is even • The smaller of the two integers is odd

22. Another proof by cases • Theorem: for all integers n, 3n2 + n + 14 is even • How could we prove this using cases? • Be careful with formatting

23. Floor and Ceiling

24. More definitions • For any real number x, the floor of x, written x, is defined as follows: • x = the unique integer n such that n ≤ x < n + 1 • For any real number x, the ceiling of x, written x, is defined as follows: • x = the unique integer n such that n – 1 < x ≤ n

25. Proofs with floor and ceiling • Prove or disprove: • x, y R, x + y = x + y • Prove or disprove: • x R, m Zx + m = x + m

26. Examples • Give the floor for each of the following values • 25/4 • 0.999 • -2.01 • Now, give the ceiling for each of the same values • If there are 4 quarts in a gallon, how many gallon jugs do you need to transport 17 quarts of werewolf blood? • Does this example use floor or ceiling?

27. Upcoming

28. Next time… • Indirect argument • Proof by contradiction • Irrationality of the square root of 2 • Infinite number of primes

29. Reminders • Turn in Assignment 2 by midnight tonight! • Keep reading Chapter 4