Counterexample

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

Definition

Definition

A counterexample is an example
that makes a statement false.

Only one counterexample is needed
to show that the statement is false.

Example 1

Example

Solution

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 2

Example

Solution

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.