Disproof by counterexample

In a nutshell

Disproof by counterexample means to show that a statement is incorrect by providing an example that proves the statement wrong.

How to disprove using a counterexample

Definition

A counterexample is an example that proves a statement to be wrong.

To disprove a statement by counterexample, it is first important to understand what you are looking for. This can be deduced by reasoning with the statement.

Example 1

Disprove the statement "the product of two prime numbers is an odd number".

The statement is saying that two prime numbers will always multiply to give an odd number - therefore, a counterexample would be finding two prime numbers that multiply to give an even number.

The product of two numbers is even if one of the numbers is even, so an even prime is needed as one of the numbers.

The only even prime is $2$ .

$2\times5=10$

$2$ and $5$ are both prime numbers, but their product is even.

This therefore counters the statement, proving it to be false.

There exists a counterexample and so the statement is false.

Example 2

Disprove the statement " $x^2>2x$ for all positive values of $x$ ".

A sufficient counterexample to this statement would be finding a positive value of $x$ that does NOT satisfy the inequality. In other words, a value of $x$ where:

$x^2\leq 2x$ 

Using trial and error with small values of $x$ :

$\begin {aligned}x&=1\Rightarrow x^2=1,2x=2 \\x&=1\Rightarrow x^2\leq 2x\end {aligned}$

$\underline{x=1}$ is a counterexample to the statement, meaning it is false.