Counterexample – Explanation and Examples

A counterexample is a specific example for which a statement is untrue.

The existence counterexample proves the statement is false, even if it is often, mostly, or almost entirely true.

Counterexamples are incredibly important for logic, which is, in turn, important because it provides the foundation for almost all mathematics.

This section covers:

• What is a Counterexample?
• How to Write a Counterexample
  

What Is a Counterexample?

A counterexample is a specific instance for which a given statement is false.

In formal logic, a statement is “false” if it is not true in every single circumstance. Even if it is usually true or sometimes true, it is false.

For this reason, finding even one case where the statement is false makes it false. Sometimes, it is possible to amend a statement to make it true by making it less general. For example, one could change a statement so that it doesn’t apply to all real numbers but just to positive numbers.

How to Write a Counterexample

Writing a counterexample requires finding a specific case when a statement is wrong and showing how the case disproves the statement.

When showing that a statement is false, it is necessary to use a specific example. As an illustration, rather than saying “this is untrue for negative numbers,” one would say, “this is untrue for $-2$.

Even though it seems like the second statement holds less weight, it is actually the proper counterexample. It is the one that achieves the goal of proving a given statement false.

It is also not necessary to include every single counterexample. This could get exhausting quickly. For example, if a statement is untrue for even numbers, it is impossible to list all of them. A single one will do.

Most statements proven false by counterexamples are conditional statements. Therefore, it is also necessary to explain how the counterexample satisfies the conditions of the antecedent but not the conditions of the consequence.

Though mathematical examples of counterexamples abound, they are also everywhere in science, philosophy, and other disciplines.

Examples

This section covers common examples of problems involving counterexamples and their step-by-step solutions.

Example 1

A friend says, “All English-speaking countries are in the Northern hemisphere.”

Convert this statement to a conditional statement. Then, find a counterexample. Is there a way to amend this statement?

Solution

The original statement easily becomes, “If it is an English-speaking country, then it is in the Northern hemisphere.”

In this case, the antecedent is “English-speaking country,” and the consequence is “Northern hemisphere.”

You happen to know, however, that there are several English-speaking countries in the Southern hemisphere. These include Australia, New Zealand, and South Africa.

Rather than saying, “there are English-speaking countries that are not in the Northern hemisphere,” however, you have to say one country specifically. Thus, you say, “Australia is not in the Northern hemisphere,” and you point it out on a map.

This disproves the statement. Since there are multiple examples, there is not an easy amendment to this statement.

Example 2

Josie says the statement, “All odd numbers under $10$ are prime is true because $7$ is less than $10$ and $7$ is prime.

Explain why providing an example does not prove a statement true, and provide a counterexample for the given statement. Can the statement be amended to be true?

Solution

A statement can be true for a million instances, but if it is false in even one case, the statement is false. Therefore, naming instances where a statement is true to prove it is true only works when there is a finite list of instances for which the antecedent is true.

That is actually the case in this example. There are only $5$ odd numbers less than $10$: $1$, $3$, $5$, $7$, $9$. Josie could prove this is true by showing that each of those is prime.

But that would not work because $9$ is not. Therefore, it makes a great counterexample. Since $9 = 3\times3$, $9$ is composite.

This statement can be amended depending on the definition of prime used. The classical definition of a number being divisible by $1$ and itself allows for $1$ to be prime, but the modern definition of a number being divisible by exactly $2$ numbers does not. Therefore, to be on the safe side, amend the statement to “if an odd number is greater than $1$ and less than $9$, then it is prime.”

It is also important to note that $2$. Is *not* a counterexample for this second statement.

Yes, $2$ is a prime, but it satisfies the consequence and not the antecedent. A proper counterexample should satisfy the antecedent but not the consequence.

Example 3

If a number is between $1^2$ and $10^2$, then its square root is between $1$ and $10$.

Why is this statement false?

Solution

This statement is false because each number has two square roots, a positive and a negative one. The negative one will be less than $1^2 = 1$.

However, the proper way to do this is to pick a random number between $1^2$ and $10^2$, preferably a perfect square. Then, show that there is a negative square root.

Pick $4$. While $2^2=4$, $(-2)^2 = 4$ too. Therefore, $-2$ is a square root of $4$, which is between $1^2 and 10^2$. Thus, the statement is untrue.

This statement, however, is easy to fix. Just change, “if a number is between $1^2$ and $10^2$, then it’s square root is between $1$ and $10$,” to “if a number is between $1^2$ and $10^2$, then the absolute value of its square root is between $1$ and $10$.” )Note that there is more than one way to reword this, so it is true.)

Example 4

There are no numbers on the interval $(1, 2)$.

Finn says there are numbers on the interval because not all numbers are whole numbers. This is not, however, a counterexample.

Can you think of a specific counterexample and a way to amend the statement?

Solution

Finn is right that not all numbers are whole numbers, so that interval is not empty. But, this is not a counterexample because it is not a specific instance.

Thus, a specific counterexample is $1.3$ or $\frac{\pi}{2}$, or $\frac{8}{7}$. Just one of these is sufficient.

A proper amendment to the statement is, “There are no whole numbers in the interval $(1, 2)$.”

Example 5

Your friend says that the statement “All winged dogs have stripes” is false. You disagree. Why?

Solution

In order to show for certain that this statement is false, you must find a specific counterexample. That is, you must find a specific instance of something that satisfies the antecedent but not the consequence.

But, there are no winged dogs. Therefore, you cannot find one that does not have stripes. For this strange reason, this, and other absurd statements, are technically true.

Another example is the statement, “all whole numbers under $100$ that are evenly divisible by $1000$ are also prime. If a number is divisible by $1000$, it is not prime by definition. But, since there are no whole numbers under $100$ that are evenly divisible by $1000$, it is impossible to show this is false with a counterexample.

Practice Problems

1. Find counterexamples to these statements about the animal kingdom.
   A. No mammals can fly
   B. If it lays eggs, then it is not a mammal
   C. If it gives birth to live young, then it is a mammal
2. “Angle measures greater than 89 degrees are obtuse.” Find a counterexample to this statement, and then amend it.
3. “The whole numbers from $90$ to $96$ are all composite.” Show that this statement is true by testing every instance.
4. Is the statement “All planets in our solar system with more than 200 moons have no rings” true or false?
5. Some countries have more than $100,000,000$ people. Explain how an example proves this statement true, not false.

Answer Key

1. A. A fruit bat is a mammal, and it can fly.
   B. A platypus lays eggs, and it is a mammal.
   C. A Great White Shark gives birth to live young, but it is not a mammal.
2. This is false. Consider an angle with a measure of $89.5$ degrees. It is still acute because it is less than a right angle. The correct statement is, “Angle measures greater than 90 degrees are obtuse.”
3. There are only $7$ whole numbers on that interval. Showing each whole number is composite only requires proving it is not prime. That is, it is not required to find all factors of the numbers. Just one other than $1$ and itself is sufficient.
   90: Divisible by 2.
   91: Divisible by 7.
   92: Divisible by 2.
   93: Divisible by 3.
   94: Divisible by 2.
   95: Divisible by 5.
   96: Divisible by 2.
4. It is impossible to prove this statement false because no planet in our solar system has more than 200 moons to the best of current scientific knowledge.
5. The United States has a population of about $330,000,000$ people, which is more than $100,000,000$. Therefore, this statement is true.