1. Week 4 - Monday

2.  What did we talk about last time?  Rational numbers

4.  An epidemic has struck the Island of Knights and Knaves  Sick Knights always lie  Sick Knaves always tell the truth  Healthy Knights and Knaves are unchanged  During the epidemic, a Nintendo Wii was stolen  There are only three possible suspects: Jacob, Karl, and Louie  They are good friends and know which one actually stole the Wii  Here is part of the trial's transcript:  Judge (to Jacob): What do you know about the theft?  Jacob: The thief is a Knave  Judge: Is he healthy or sick?  Jacob: He is healthy  Judge( to Karl): What do you know about Jacob?  Karl: Jacob is a Knave.  Judge: Healthy or sick?  Karl: Jacob is sick.  The judge thought a while and then asked Louie if he was the thief. Based on his yes or no answer, the judge decided who stole the Wii.  Who was the thief?

6.  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"

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

8.  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?

9.  Prove that for all integers a, b, and c, if a | b and b | c, then a | c  Steps:  Rewrite the claim in formal notation  Write Proof:  State your premises  Justify every line you infer from the premises  Write QED after you have demonstrated the conclusion

10.  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?

11.  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

12.  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

14.  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

15.  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

16.  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?

17.  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

18.  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

19.  Theorem: for all integers n, 3n 2 + n + 14 is even  How could we prove this using cases?  Be careful with formatting

21.  Indirect proof  Classic results:  Irrationality of the square root of 2  Infinitude of primes

22.  Exam 1 is next Monday  Review is Friday  Read sections 4.6 and 4.7