Counterexample

Counterexample

How to solve counterexample problems: meaning, examples, and their solutions.

What is it?

A counterexample is an example that makes a statement to be false.

A counterexample is an example
that makes a statement to be false.

Only one counterexample is needed
to prove the statement to be false.

Example 1

Find a counterexample for the given statement: If x is a prime number, then x is an odd number.

If x = 2,
then p is true (2 is a prime number)
and q is false (2 is not an odd number).
So pq is false.

Conditional statement: truth value

So the counterexample is x = 2.

Example 2

Find a counterexample for the given statement: Three points determine a triangle.

If three points are on a same line,
then those three points
cannot determine a triangle.

So it makes the given statement to be false.

So the counterexample is
'three points on a same line'.

To make this statement to be true,
it should be like this:
three non-linear points determine a triangle.