Proofs. 29th January 2014

Transcription

1 Proofs 29th January 2014

2 Housekeeping Your solutions to Problem Sheet 2 are due today at the start of class. Please make sure you have your name on them and that you put them in the correct pile! Don t forget to prepare your journal entry for Friday. Problem Sheet 3 has been posted to Blackboard. Some of the problems ask you to do questions listed in the Velleman reading on Blackboard.

3 Intro Last time we looked at Euclid s system of geometry. We saw that he wanted to show geometrical theorems were known with certainty by deriving them from a limited number of axioms, i.e. by proving them. We also discussed other roles that proofs can play in mathematics. Now, we re going to do some proofs!

4 A Few Remarks Writing proofs can be hard! There s no general recipe for solving every problem. But don t give up! If an approach you try doesn t work, don t just throw it away examining it might help you identify a new strategy.

5 Warm Up: Pythagorean Theorem Theorem If we have a right angled triangle with hypotenuse of length c and legs of length a and b then a 2 + b 2 = c 2

6 Proof of Pythagorean Theorem

7 Proof of Pythagorean Theorem This proof is from Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures by James Robert Brown Area of big square is (a + b) 2 = a 2 + b 2 + 2ab. Area of smaller square is c 2. c 2 = (a + b) 2 4( 1 2 ab) = (a + b) 2 2ab = a 2 + b 2

8 Proof Strategies In the reading from Velleman you saw how to prove mathematical statements of the form if p then q. The above proof of the Pythagorean Theorem is of that form. Let s look at example that illustrates a different kind of proof strategy.

9 2 is irrational What does it mean for a number n to be rational? n can be represented as p q, p, q are integers, q 0. Moreover, we can choose p and q so that they are coprime, i.e. there are no positive integers other than 1 which evenly divide both p and q.

10 2 is irrational A number n is irrational if it is not rational.

11 2 is irrational We ll need to use the following fact: For any integer n, if 2 divides n 2 then 2 divides n. This holds because 2 is prime. For any prime p and any integers a, b, if p divides ab then p divides a or p divides b.

12 Proof Strategy How might we prove the claim that 2 is irrational, i.e. it is not rational? We can assume that 2 is rational and show that this leads to a contradiction. In other words, from the assumption that 2 is rational we obtain both a statement and its negation. This shows that it must be false that 2 is rational, i.e. it must be irrational. This is an example of proof by contradiction.

13 Proof that 2 is irrational Proof. Suppose that 2 is rational. Then there are coprime integers p, q with q 0 such that 2 = p q Squaring both sides, we get 2 = p2. q 2 Multiply both sides by q 2 to obtain 2q 2 = p 2. Thus p 2 is even and so p must be even. Thus p = 2k for some integer k. Substituting this back in to the above we obtain 2q 2 = 4k 2. Thus q 2 = 2k 2. Hence q 2 is even and thus q is even. But 2 divides p and q and 2 1 so p, q are not coprime. This contradicts our previous assumption. Hence 2 is irrational.

14 Practice! We ve seen some different proof strategies now, so let s try to put them into action! I d like you all to think about how to prove the results on the next few slides, and try to prove them yourself. It will be helpful to have a pen and paper ready!

15 Prove It Yourself! The following is an exercise from Velleman s textbook How To Solve It: Suppose a, b, c, and d are real numbers, 0 < a < b, and d > 0. Prove that if ac bd then c > d. Proof. Suppose that c > d is false, i.e. c d. Then as a > 0, multiplying both sides of c d by a, we have ac ad. Also as d > 0 if we multiply both sides of 0 < a < b by d we obtain 0 < ad < bd. Combining both of these inequalities we obtain ac < bd. Thus if c d then ac < bd or equivalently if ac bd then c > d.

16 Prove It Yourself 2! The following is an exercise from Velleman s textbook How To Solve It: Suppose that y + x = 2y x, and x and y are not both zero. Prove that y 0. Proof. y + x = 2y x, and x and y are not both zero. For a contradiction, assume that y = 0. Then 0 + x = 2 0 x = x Rearranging gives 2x = 0, which implies x = 0. This means that x and y are both zero. But this contradicts the fact that x and y are not both zero. Hence if y + x = 2y x, and x and y are not both zero then y 0.

17 Bad Proofs! Here s a bad proof from Velleman s How To Solve it. What s wrong with it? Theorem If x, y are real numbers and x + y = 10 then x 3 and y 8. Proof. Suppose the conclusion of the theorem is false. Then x = 3 and y = 8. But then x + y = 11, which contradicts the given information that x + y = 10. Therefore the conclusion must be true.

18 Bad Proofs! Theorem If x, y are real numbers and x + y = 10 then x 3 and y 8. Proof. Suppose the conclusion of the theorem is false. Then x = 3 and y = 8. But then x + y = 11, which contradicts the given information that x + y = 10. Therefore the conclusion must be true. The negation of x 3 and x 8 is not x = 3 and y = 8. Why? For x 3 and x 8 to be false, just one of the conjuncts has to be false, not both. So the negation is x = 3 or y = 8.

19 Bad Proofs! Theorem If x, y are real numbers and x + y = 10 then x 3 and y 8. This theorem is not a theorem at all! Come up with a counter example! x = 3, y = 7 is a counter example. x = 2, y = 8 is a counter example.

20 Sums of Natural Numbers What s the sum of the first 100 natural numbers? =?? This question was (supposedly) set by a teacher to keep their class quiet. But they had a young Gauss in their class! According to the story, Gauss solved it almost instantly. How did he do it?

21 Sums of Natural Numbers =?? The key is to consider summing this twice, then divide by two.

22 Sums of Natural Numbers =?? Let S = Then 2S = Notice that each pair of terms sums to 101. As there are 100 of these terms, 2S = = 10, 100. Thus S = 10,100 2 = 5, 050.

23 Sums of Natural Numbers This approach generalizes! What will the sum of the first n numbers be? S(n) = n =??

24 Sums of Natural Numbers We can show that the sum of the first n natural numbers is n2 2 + n 2. Let S(n) = n. Then 2S(n) = n + n + (n 1) So 2S(n) = n(n + 1) = n 2 + n and hence S(n) = n2 2 + n 2.

25 Philosophical Questions What roles do pictures and diagrams play in proofs? Can a picture by itself ever be a proof?

26 Next Time... The reading for next week is a little different to what we ve seen so far! Try to work out what the arguments are (which is harder than it might first appear!) and whether they are valid or sound.