Proof Y13

Question 1

What do you NEED to do when proving by contradiction?

Answer 1

assume that the opposite to the statement is true
then show this would lead to a contradiction so that the original statement is true

Question 2

Prove that √2 is irrational

Answer 2

Assume that √2 is rational and can be written in the form a/b where a and b have no common factors
√2 = a/b
2 = a^2 / b^2
a^2 = 2b^2
a^2 is a multiple of 2 –> a is a multiple of 2 also
let a =2m, a^2=4m^2, 4m^2=2b^2, b^2=2m^2
–> b^2 is a multiple of 2, so b is a multiple of 2
since a and b are both multiples of 2 they both have a common factor
–>contradicts original assumption
–>√2 is irrational

Question 3

Prove by contradiction that there are an infinite number of primes

Answer 3

assume there is a finite number of n primes
let P = P1 x P2 x P3 x … x Pn + 1
If you divide P by any of P1, P2, P3, .. Pn, you will you have a remainder of 1
∴ none of the Primes are factors of p
∴ P must be prime
this contradicts the original statement
∴ there are infinitely many primes

Question 4

Prove that the sum of a rational number and an irrational number is always irrational

Answer 4

assume the sum is always rational
let a/b be a rational number
let x be irrational
∴ a/b + x = c/d where c/d is a rational number
x = c/d - a/b
x = bc - ad / bd
∴ x is a rational number
∴ contradicts original statement
∴ the sum of a rational number and an irrational number is always irrational