All About Proofs Lecture 2: Sep 5 (based on slides in MIT 6.042)
c b a Pythagorean theorem Familiar? Obvious? Yes! No!
A Cool Proof c b a Rearrange into: (i) a cc square, and then (ii) an aa & a bb square
A Cool Proof c b-a c c a b c
b-a b-a A Cool Proof c b a
A Cool Proof a b a a b-a b 74 proofs in http://www.cut-the-knot.org/pythagoras/index.shtml
1 1 10 1 1 1 11 Getting Rich By Diagram 1 1 11 1 10
Getting Rich By Diagram 1 1 10 11 1 1 10 11 Profit! 1 1
1 1 1 1 Getting Rich By Diagram The bug: are not right triangles! The top and bottom line of the “rectangle” is not straight! 10 1
Evidence vs. Proof Let p(n) ::= n2 + n + 41. Claim: p(n) is a prime number for all nonnegative integer n. prime p(0) = 41 Check: prime p(1) = 43 p(2) = 47 prime prime p(3) = 53 looking good! prime p(20) = 461 p(39) = 1601 prime enough already!
Evidence vs. Proof Let p(n) ::= n2 + n + 41. Claim: p(n) is a prime number for all nonnegative integer n. Well, it is clear that the claim is true p(40) = 1681 is not a prime number. Why? But NO! p(40) = 402 + 40 + 41 = 402 + 2x40 + 1 = (40+1)2
Evidence vs. Proof Let p(n) ::= n2 + n + 41. Actually, p(n-1) = p(-n), so if we consider f(n) = p(n - 40) = n2 + 79n + 1601 f(n) is a prime for x=0,1,2,…,79 This is the current champion (the function n2 - 2999n + 224851 also yields 80 consecutive primes from 1460 to 1539).
More Claims is a prime number for all nonnegative n. Claim: f(0)=3, f(1)=5, f(2)=17, f(3)=257, f(4)=65537, Fermat number f(5)=4294967297 The first 49 Fermat numbers haven been checked, but except for those up to f(4) every one is composite (not prime)! Euler conjecture: has no solution for a,b,c,d positive integers. Open for 218 years, until Noam Elkies found
Even More Claims Fermat (1637): If an integer n is greater than 2, then the equation an + bn = cn has no solutions in non-zero integers a, b, and c. Andrew Wiles (1994): prove it using “elliptic curves”. http://en.wikipedia.org/wiki/Fermat's_last_theorem has no solutions in non-zero integers a, b, and c. Claim: False. But smallest counterexample has more than 1000 digits. Goldbach’s conjecture: Every even number is the sum of two prime numbers.
Proving an Implication Goal: If P, then Q. (P implies Q) Method 1: Write assume P, then show that Q logically follows. Claim: If , then Reasoning: When x=0, it is true. When x grows, 4x grows faster than x3 in that range. Proof: When (see page 19 of the book)
Proving an Implication Goal: If P, then Q. (P implies Q) Method 1: Write assume P, then show that Q logically follows. Claim: If r is irrational, then √r is irrational. How to begin with? What if I prove “If √r is rational, then r is rational”, is it equivalent? Yes, this is equivalent; proving “if P, then Q” is equivalent to proving “if not Q, then not P”.
Proving an Implication Goal: If P, then Q. (P implies Q) Method 2: Prove the contrapositive, i.e. prove “not Q implies not P”. Claim: If r is irrational, then √r is irrational. Proof: We shall prove the contrapositive – if √r is rational, then r is rational. Since √r is rational, √r = a/b for some integers a,b. So r = a2/b2. Since a,b are integers, a2,b2 are integers. Therefore, r is rational. (Q.E.D.) "which was to be demonstrated", or “quite easily done”.
Proving an “if and only if” Goal: Prove that two statements P and Q are “logically equivalent”, that is, one holds if and only if the other holds. Example: An integer is a multiple of 3 if and only if the sum of its digits is a multiple of 3. Method 1: Prove P implies Q and Q implies P. Method 1’: Prove P implies Q and not P implies not Q. Method 2: Construct a chain of if and only if statement. (see page 21 of the book)
Propositional (Boolean) Logic Proposition is either True or False True 2 + 2 = 4 Examples: False 3 x 3 = 8 787009911 is a prime Non-examples: Hello. How are you?
P P Q Q P Q P Q T T T T T T T T F F T F F F T T T F F F F F F F Logic Operators exclusive-or Connecting propositions. coffee “or” tea
P P Q Q P Q P Q T T T T T T T T F F F F F F T T F T F F F F T T Note: P Q is equivalent to (P Q) (Q P) Logic Operators Convention: if we don’t say anything wrong, then it is true. Note: P Q is equivalent to (P Q) ( P Q)
Math vs English Parent: if you don’t clean your room, then you can’t watch a DVD. C D This sentence says So In real life it also means Mathematician: if a function is not continuous, then it is not differentiable. This sentence says But of course it doesn’t mean
Logical Deduction From: P implies Q, Q implies R To: P implies R antecedents conclusion Definition: A rule issound if the conclusion is true whenever all antecedents are true.
More Examples sound sound READ CHAPTER 1!! sound unsound sound