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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!

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.

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:

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.

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.

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.

Introduction to Proofs Flashcards

Introduction to Proofs Practice Tests

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