Proof

Type 1: Disproof by Counterexample

Disproof by counter example is a way of disproving a statement by providing one example which doesn’t work for the statement.

Example: Hernan claims “if you square a number and add 1, the result is a prime number”. Find a counterexample to prove her statement wrong.

To make life easy, we’ll start with the smaller numbers.

1^2+1=1+1=2,\text{ which is prime.}

2^2+1=4+1=5,\text{ which is prime.}

Now, at this point it might seem like her statement is true, but if we try the next square number:

3^2+1=9+1=10,\text{ which is not prime.}

This is a counterexample for her statement, so we have proved it to be false.

Type 2: Algebraic Equivalent

Algebraic equivalent proofs will present you with a two sided equation, usually with a \equiv rather then an =. The task is to play around with one side until it looks like the other.

Example: 3(n + 3)(n − 1)− 3(1− n) \equiv (3n − 3)(n + 4)

Here we need to expand out all the brackets and collect the resulting like terms.

\begin{aligned}3(n + 3)(n − 1)− 3(1− n) &\equiv (3n − 3)(n + 4) \\ 3(n^2+2n-3)-3+3n &\equiv 3n^2 +12n-3n-12 \\ 3n^2 +6n-9-3+3n &\equiv 3n^2 +12n-3n-12\\ 3n^2 +9n-12 &\equiv 3n^2 +9n-12 \end{aligned}

We can see now that the two sides of the equation are equivalent (\equiv)

Type 4: Algebraic Proof

Algebraic proofs involve constructing an algebraic expression to match the statement, then proving or disproving the statement with this expression.

When constructing algebraic proof, we can associate common statements with their equivalent algebraic expressions.

The list below summarises statements that frequently occur. 

• Even numbers \rightarrow 2n
• Odd numbers \rightarrow 2n+1
• Consecutive numbers \rightarrow n \text{ and } n+1
• Consecutive even numbers \rightarrow 2n \text{ and } 2n+2
• Consecutive odd numbers \rightarrow 2n+1 \text{ and } 2n+3
• Proving a statement is always positive or a square number – the expression would need to be simplified to (...)^2
• Proving a statement is a multiple of x or divisible by x – the expression would need to be factorised to the form x(...)

This topic can get very challenging, but a simple example is shown below.

Example: Prove algebraically that the sum of two consecutive numbers is odd.

Step 1: Form the algebraic expression.

We know a number can be represented by n, so the next number would be (n+1).

n + (n+1)

Step 2: Expand / simplify the expression as much as possible.

n + n+1 = 2n+1

Step 3: Use the expression to prove the statement.

For this example, this is easy as the expression is already in the correct format.

 2n+1 =  Odd number

Type 5: Proofs with indices

Questions involving indices will require a good knowledge of your indices rules.

Example: Prove that the difference between 24^{12} and 15^{10} is a multiple of 3

Step 1: Write the expression to find the difference

24^{12}-  15^{10}

Step 2: Take a out a factor of 3 from each term.

\begin{aligned}(24\times24^{11}) &-  (15\times15^9) \\ (3\times8\times24^{11}) &- (3\times5\times15^9)\end{aligned}

Step 3: Factorise by 3

(3\times8\times24^{11}) - (3\times5\times15^9) = 3([8\times24^{11}] - [5\times15^9])

We know that 3\times(\text{any integer}) is a multiple of 3 therefore,

3([8\times24^{11}] - [5\times15^9]) is a multiple of 3

Type 6: Proofs with Primes

A common question involves proving that an expression isn’t prime.

Example: Show that 5^{89} - 401 is not a prime number.

We know Odd \times Odd = Odd, therefore,

5^{89} is Odd

We know Odd - Odd=Even

5^{89} - 401  = Even

Even numbers cannot be prime.