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 ⇒ x2 = y 2 = xy (1) ⇒ x2 − y 2 = xy − y 2 (2) ⇒ (x − y)(x + y) = (x − y)y (3) ⇒ x + y = y. (4) As x = y, x + y = y ⇒ 2y = 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 = a2/b2 (7) 2b2 = a2. (8) So a = 2c This gives 2b2 = (2c)2 = 4c2 ⇒ b2 = 2c2. (9) But if b is even then b2 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 = a2 b2 ⇒ 2b2 = a2 meaning that a2 is even, and so a must be even as well. This means that we can write a = 2c for some c ∈ Z. This gives 2b2 = (2c)2 = 4c2 ⇒ b2 = 2c2. 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 n2 + n + 11 is a prime number for all positive integer values of n. Question 3. Prove that, for all n ∈ N, ∞ xne−xdx = n!