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CS322

Week 3 - Wednesday. CS322. Last time. What did we talk about last time? Basic number theory definitions Even and odd Prime and composite Proving existential statements Disproving universal statements. Questions?. Logical warmup.

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CS322

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  1. Week 3 - Wednesday CS322

  2. Last time • What did we talk about last time? • Basic number theory definitions • Even and odd • Prime and composite • Proving existential statements • Disproving universal statements

  3. Questions?

  4. Logical warmup • By their nature, manticores lie every Monday, Tuesday and Wednesday and the other days speak the truth • However, unicorns lie on Thursdays, Fridays and Saturdays and speak the truth the other days of the week • A manticore and a unicorn meet and have the following conversation:Manticore: Yesterday I was lying.Unicorn: So was I. • On which day did they meet?

  5. Proving Universal Statements

  6. Method of exhaustion • If the domain is finite, try every possible value. • Example: • x  Z+,if 4 ≤ x ≤ 20 and x is even, x can be written as the sum of two prime numbers • Is this familiar to anyone? • Goldbach's Conjecture proposes that this is true for all even integers greater than 2

  7. Generalizing from the generic particular • Pick some specific (but arbitrary) element from the domain • Show that the property holds for that element, just because of that properties that any such element must have • Thus, it must be true for all elements with the property • Example: x  Z,if x is even, then x + 1 is odd

  8. Direct proof • Direct proof actually uses the method of generalizing from a generic particular, following these steps: • Express the statement to be proved in the form x  D,if P(x) then Q(x) • Suppose that x is some specific (but arbitrarily chosen) element of D for which P(x) is true • Show that the conclusion Q(x) is true by using definitions, other theorems, and the rules for logical inference

  9. Proof formatting • Write Claim: and then the statement of the theorem • Start your proof with Proof: • Define everything • Write a justification next to every line • Put QED at the end of your proof • Quod erat demonstrandum: "that which was to be shown"

  10. Direct proof example • Prove the sum of any two odd integers is even.

  11. Common mistakes • Arguing from examples • Goldbach's conjecture is not a proof, though shown for numbers up to 1018 • Using the same letter to mean two different things • m = 2k + 1 and n = 2k + 1 • Jumping to a conclusion • Skipping steps • Begging the question • Assuming the conclusion • Misuse of the word "if • A more minor problem, but a premise should not be invoked with "if"

  12. Disproving an existential statement • Flipmode is the squad • You just negate the statement and then prove the resulting universal statement

  13. Rational Numbers Student Lecture

  14. Rational Numbers

  15. 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

  16. Examples • Give an example of a rational number • Given an example of an irrational number • Is 10/3 rational? • Is 0.281 rational? • Is 0.12121212… (the digits 12 repeat forever) rational? • Is 0 rational? • Is 3/0 rational? • Is 3/0 irrational? • If m and n are integers and neither is zero, is (m + n)/mn rational?

  17. Prove the following: • Every integer is a rational number • The sum of any two rational numbers is rational • The product of any two rational numbers is a rational number

  18. 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

  19. Divisibility

  20. 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"

  21. Examples • Is 37 divisible by 3? • Is -7 a factor of 7? • Does 6 | 256? • Is 0 a multiple of 45?

  22. More on divisors • If a,b Z and a | b, is a ≤ b? • Not necessarily! • But, if a,b Z+ and a | b, then a ≤ b • Which integers divide 1? • If a,b Z, is 3a + 3b divisible by 3? • If k,m  Z, is 10km divisible by 5?

  23. 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

  24. Divisibility by a prime • Prove that every integer n > 1 is divisible by a prime • 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 • This one is a little more problematic • We aren't really ready to prove something like this yet

  25. 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?

  26. 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

  27. 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

  28. Upcoming

  29. Next time… • Quotient remainder theorem • Proof by cases

  30. Reminders • Keep reading Chapter 4 • Work on Assignment 2 for Friday

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