Week 3 - Wednesday

 What did we talk about last time?  Basic number theory definitions  Even and odd  Prime and composite  Proving existential statements  Disproving universal statements

 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?

 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

 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

 Direct proof actually uses the method of generalizing from a generic particular, following these steps: 1. Express the statement to be proved in the form  x  D, if P(x) then Q(x) 2. Suppose that x is some specific (but arbitrarily chosen) element of D for which P(x) is true 3. Show that the conclusion Q(x) is true by using definitions, other theorems, and the rules for logical inference

 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"

 Prove the sum of any two odd integers is even.

 Arguing from examples  Goldbach's conjecture is not a proof, though shown for numbers up to  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"

 Flipmode is the squad  You just negate the statement and then prove the resulting universal statement

Student Lecture

 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

 Give an example of a rational number  Given an example of an irrational number  Is 10/3 rational?  Is rational?  Is … (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?

 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

 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: 1. The sum, product, and difference of any two even integers is even 2. The sum and difference of any two odd integers are even 3. The product of any two odd integers is odd 4. The product of any even integer and any odd integer is even 5. 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 (a 2 + b 2 + 1)/2 is an integer

 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"

 Is 37 divisible by 3?  Is -7 a factor of 7?  Does 6 | 256?  Is 0 a multiple of 45?

 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?

 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

 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

 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?

 For any integer n > 1, there exist a positive integer k, distinct prime numbers p 1, p 2, …, p k, and positive integers e 1, e 2, …, e k 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

 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

 Quotient remainder theorem  Proof by cases

 Keep reading Chapter 4  Work on Assignment 2 for Friday