Counter Example

In the world of math, we like to create our own special words. One of these words is "counterexample," and its meaning is fairly obvious. Becoming more familiar with these terms helps us approach the field of mathematics with greater confidence. The next time someone asks you for a counterexample, you'll be ready with an answer!

The definition of a counterexample

A counterexample is an example that shows everyone that a general statement is false. You've probably given plenty of counterexamples in your life.

For example, someone might have said: "Everyone knows that you can't use your hands in soccer," to which you might respond: "But the goalies are allowed to use their hands!"

You've just provided a counterexample.

A counterexample in geometry

Of course, we're dealing with much more specific arguments when we use counterexamples in math.

One such example involved geometry. You might hear someone say:

"Every quadrilateral has at least two congruent sides."

To which you might respond with the counterexample:

"A scalene quadrilateral does not have at least two congruent sides."

You might also draw your scalene quadrilateral to further illustrate your point:

A counterexample of a conditional statement

You may remember that conditional statements follow the following basic structure:

"If p, then q."

So how exactly do you give a counterexample to a conditional statement?

Let's use this example:

For all real valued numbers a and b, there is a unique value x such that if $\left(a\right)\left(b\right)=x$ , then $\left(a\right)=\frac{x}{b}$ .

Can we prove that this is not true for all real numbers substituted for a, b?

If we let $a=1$ and $b=0$ ,

Then $1\left(0\right)=0$ , but $1\ne \frac{0}{0}$ since division by zero is undefined.

Therefore, we have just provided a counterexample for this conditional statement.

What's the difference between a counterexample and a regular example?

So why can't we just call counterexamples "examples" and make everything easier for everyone?

Well, counterexamples are very specific types of examples. We might provide an example to support our statements -- but counterexamples always refute or disprove statements.

If we want to provide examples to support our argument that mammals can't lay eggs, we might point to animals like the deer, the cow, and the human.

But if someone wanted to provide a counterexample to that statement, they could point to the platypus.

In short, counterexamples refute arguments while examples support them.

No amount of examples will ever prove a claim, only lend support, but it only takes a single counterexample to disprove it entirely.

Famous counterexamples in history

As you might have guessed, mathematicians and great thinkers have come up with all kinds of interesting counterexamples over the years.

Euler's sum of power conjecture was disproved with counterexamples. This conjecture asserted that to represent a given number as a sum of n kth powers, at least n terms are needed. In 1966, L. J. Lander and T. R. Parkin provided a valid counterexample involving $n=5$ and $k=4$ .

Witsenhausen provided a counterexample that showed that it's not always true that a quadratic loss function and a linear equation of evolution of the state variable lead to optimal control laws that are linear. This is known as Witsenhausen's counterexample in control theory.

Counterexamples are also very common in philosophy. In the dialogue "Euthyphro" by Plato, Socrates provides a counterexample to Euthyphro's definition of piety. Euthyphro suggests that what is pious is what is pleasing to the gods. Socrates points out that the gods often disagree, so what is pleasing to one god might not be pleasing to another, thus refuting Euthyphro's definition.

Topics related to the Counter Example

Symbolic Logic

Axiom

Conjunction

Flashcards covering the Counter Example

Symbolic Logic Flashcards

Introduction to Proofs Flashcards

Practice tests covering the Counter Example

Introduction to Proofs Practice Tests

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