# Counterexample

How to find the counterexample of the given statement: definition, 2 examples, and their solutions.

## Definition

A counterexample is an example

that makes a statement false.

Only one counterexample is needed

to show that the statement is false.

## Example

The given statement is a conditional statement.

p: x is a prime number.

q: x is an odd number.

Recall that

if p is true and q is false,

then p → q is false.

Conditional Statement: Truth Value

So, to find the counterexample,

find the x that makes

p true (= x is a prime number)

and q false (= x is not an odd number).

Set x = 2.

If x = 2,

p: 2 is a prime number.

This is true.

If x = 2,

q: 2 is an odd number.

This is false.

If x = 2,

p is true

and q is false.

Then p → q is false.

x = 2 makes

the given conditional statement false.

So x = 2 is the counterexample

of the given statement.

So [x = 2] is the answer.

## Example

Find the case

that makes the given statement false.

Think of the case when

three points are on the same line.

As you can see,

these three points

cannot determine a triangle.

So the given statement is false.

[Three points on the same line]

makes the given statement false.

So [three points on the same line]

is the counterexample

of the given statement.

So [three points on the same line]

is the answer.