Proof and mathematical communication

Question 1

What is the main objective of proof by contradiction

Answer 1

The attempt to prove the opposite statement to the conjectured statement and show the the result creates and impossible contradiction which results in the conjectured statement being true

Question 2

Use proof by contradiction to prove there are an infinite number or primes.

Answer 2

Assume there is a largest prime number P and hence there is a finite number of primes.

Constuct a unique number N such that it is a product of all the primes up to and including P.

If we consider N+1, This is one greater than a number divisible by all the primes up to and incluiding P so it cannot be divisible by any of the primes up to and including P

Therefore N+1 is either prime or is Divisible by primes larger than P

as a result, a Prime larger than P has been discovered.

Contradiction,

There are an infinite number of primes

Question 3

What are the 2 areas of a proof which need to be criticised when looking at a proof

Answer 3

Arithmetic and logic