A mathematical proof starts with a set of assumptions
Then it follows a series of logical steps
And reaches a conclusion
For single maths you are expected to be familiar with proof by deduction, proof by exhaustion, disproof by counter example and proof by contradiction.
A proof by deduction demonstrates that the truth of the conclusion follows logically from the assumptions.
For example: prove that the sum of two odd numbers is always an even number.
Any odd number can be written as (2n + 1)
where "n" is a whole number.
Any even number can be written as (2n)
where "n" is a whole number.
Assume that we have 2 odd numbers A and B
where A = (2a + 1)
B = (2b + 1)
and "a" and "b" are whole numbers
It follows that:
A + B = (2a + 1) + (2b + 1)
= 2a + 1 + 2b + 1
= 2a + 2b + 2
= 2(a + b + 1)
= 2k
As "k" is a whole number (a + b + 1)
we have shown than A + B is an even number
So we have shown by deduction that the sum of 2 odd numbers must be an even number.
When a statement cannot easily be proved for all cases in one go, it can instead be proved by considering all the partial cases one at a time.
This can be very time consuming if there are a lot of cases, so proof by exhaustion, sometimes called "brute force", is considered inelegant.
For example: prove that any perfect cube must be a multiple of 9 or one away from a multiple of 9.
Consider 3 cases for the number to be cubed:
n is a multiple of 3
n is one more than a multiple of 3
n is one less than a multiple of 3
This covers all whole numbers
case 1:
n = 3k, where k is a whole number
n3 = (3k)3
= 27k3
= 9(3k3)
which is a multiple of 9
case 2:
n = 3k + 1, where k is a whole number
n3 = (3k + 1)3
= 27k3 + 27k2 + 9k + 1
= 9(3k3 + 3k2 + k) + 1
which is a multiple of 9 plus one
case 3:
n = 3k - 1, where k is a whole number
n3 = (3k - 1)3
= 27k3 - 27k2 + 9k - 1
= 9(3k3 - 3k2 + k) - 1
which is a multiple of 9 minus one
So by splitting the proof into 3 parts, showing that those parts cover all cases, and proving each part we have shown that all perfect cubes are at most one away from a multiple of 9.
To disprove something means to show that a statement is false. To prove that something is true you have to show that it is so for all values, but to disprove it you only need to find one value, a "counter example", where the statement is false.
For example: disprove that the square of any odd number greater than 4 is one more than a multiple of 24.
Consider the following values: 52 = 25 = 1 x 24 + 1 72 = 49 = 2 x 24 + 1 92 = 81 = 3 x 24 + 9 Hence 92 is not one more than a multiple of 24
So we have disproved that the square of and odd number greater than 4 is one more than a multiple of 24 by finding a counter example of 9.
(try to prove that the square of any prime number greater than 4 is one more than a multiple of 24)
Sometimes it is easier to prove something by showing that the opposite must be false. This is proof by contradiction - you start by saying that the opposite of what you want to prove is true, then you show by deduction that this must be false, which proves that your original is true.
Two proofs by contradiction that you are expected to know are (a) proof that √2 is irrational; and (b) proof that there are an infinite number of primes.
To show that √2 is irrational
First assume the opposite, √2 is rational
Then we can write:
√2 = a / b
Where a and b are whole numbers
with no common factors.
√2 = a / b
2 = a2 / b2
a2 = 2b2
Which means that a2 is even
and therefore "a" is even.
If we write a = 2k, then:
2b2 = (2k)2
2b2 = 4k2
b2 = 2k2
Which means that b2 is even
and therefore "b" is even.
But we have shown that "a" and "b" are both even
when we stated that they had no common factor.
This is a contradiction
Therefore the assumption √2 is rational is false
Which proves that √2 is irrational
and
To show that there are infinite primes First assume the opposite, there are finite primes Then there must be a biggest prime number, which we will call "N". Consider the number P which is the product of all the primes. P = 2 x 3 x 5 x 7 x 11 x .... x N And consider the number Q Q = P + 1 What is Q? If you divide it by any prime then you get the remainder 1. So it has no factors other than 1 and itself... which makes it prime. But Q > N which was the biggest prime This is a contradiction Therefore the assumption of finite primes is false Which proves that there are an infinite number of prime numbers.