An Introduction to Proofs Sam Brzezicki, Carlisle King and Madeleine - PowerPoint PPT Presentation

An Introduction to Proofs Sam Brzezicki, Carlisle King and Madeleine Whybrow 7th October 2016

What is a proof? ◮ “A chain of reasoning using rules of inference, ultimately based on a set of axioms, that lead to a conclusions”. (Penguin Dictionary of Mathematics) ◮ “A watertight logical argument which shows with complete mathematical certainty that the result is correct.” (Sam) ◮ “A way of showing that something is true.” (An engineering friend)

Why are they important? ◮ We want to be sure that we’re sure. ◮ Because we want to learn from it.

How to write a proof 1. Write out what you have been given. 2. Ask yourself questions as you move through. 3. Make sure you go step-by-step. 4. Signify when you’ve reached the end! 5. Go back and think about your proof.

Different types of proofs ◮ Direct proof- combining axioms, definitions and known results. ◮ Proof by induction. ◮ Proof by contradiction - assume a statement is false and find a logical contradicition. ◮ Proof by exhaustion - divide into cases. ◮ Proof by counter-example - for showing a statement is false.

Example of a flawed proof Suppose that x , y ∈ R \{ 0 } and that x = y then x = y ⇒ x 2 = y 2 = xy (1) ⇒ x 2 − y 2 = xy − y 2 (2) ⇒ ( x − y )( x + y ) = ( x − y ) y (3) ⇒ x + y = y . (4) As x = y , x + y = y ⇒ 2 y = y (5) ⇒ 2 = 1 . (6)

What is a good proof? √ Question Prove that 2 is irrational. Solution Suppose that √ 2 = a b . Squaring both sides, we have 2 = a 2 / b 2 (7) 2 b 2 = a 2 . (8) So a = 2 c This gives 2 b 2 = (2 c ) 2 = 4 c 2 ⇒ b 2 = 2 c 2 . (9) But if b is even then b 2 is too. So both a and b are even, and hence √ share a common divisor. Hence 2 / ∈ Q .

What is a good proof? √ Question Proof that 2 is irrational. √ Solution Suppose for contradiction that 2 ∈ Q . Then we can write √ 2 = a b where a , b ∈ Z . Without loss of generality, we can assume that a and b share no common divisor. Squaring both sides, we have 2 = a 2 b 2 ⇒ 2 b 2 = a 2 meaning that a 2 is even, and so a must be even as well. This means that we can write a = 2 c for some c ∈ Z . This gives 2 b 2 = (2 c ) 2 = 4 c 2 ⇒ b 2 = 2 c 2 . This means that a and b are both even. However, this contradicts our assumption that a and b share no common divisors. Thus our initial √ assumption must be false and 2 / ∈ Q .

Exercises Question 1 . In the world famous East Grinstead Zoo, the elephants and the tigers all live in the same enclosure. Their relationships obey the following rules: 1. There is at least one elephant. 2. For any two different tigers, there is exactly one elephant that likes both of them. 3. Each elephant likes at least two tigers. 4. For any elephant, there is at least one tiger which it does not like. Prove the following statements: (a) There are at least three tigers. (b) If there are exactly 3 tigers, then there are also exactly 3 elephants. (c) If there are exactly 4 tigers, then there are either 4 or 6 elephants.

Exercises Question 2 . Prove or disprove the following: The value of n 2 + n + 11 is a prime number for all positive integer values of n . Question 3 . Prove that, for all n ∈ N , � ∞ x n e − x d x = n ! 0