Our favorite word in mathematics is all.

We love to make statements that all of the things that match some description, have some characteristic. These can be very difficult statements to prove true, which is why we value them so much.

Of course, some of our statements will be true and some of our statements will be false.

• When the statement is true, we would like to explain how we know it is true.
• When the statement is false, we would like to explain how we know it is false.

When dealing with the word all, it is easier to show that a false statement is false than it is to show a true statement is true.

All, Every, Each

When a statement makes a claim about all of the items in a set and we think the statement is false, then a counterexample is our explanation that the statement is false.

Counterexamples

Here is the complete graph of the function \(G(x)\).

Statements about \(G(x)\)

• \(G(x) < 0\) for all values of \(x\) in the domain.
• \(G(x) < 10\) for all values of \(x\) in the domain.
• \(G(x) > 4\) for each value of \(x\) in the domain such that \(x > 0\).
• \(G(x) < 2\) for every value of \(x\) in the domain such that \(x < 0\).

These are all statements presented through the function values of \(G\). They are either true of false. If we believe a statement is false, then we are going to provide a counterexample.

Counterexamples for statements about functions are DOMAIN numbers where the statement is false.

This follows the general flow of discussions around function analysis. Questions are posed about the function values (what) and their answers are domain numbers (where).

Let’s consider the statements above one at a time.

\(\blacktriangleright \) \(G(x) < 0\) for all values of \(x\) in the domain.

This statement is false. Our counterexample is \(2\).

\(G(2) \approx 1 > 0\). The number \(2\) is just one number in the domain. The original statement used the word all and that means that every number in the domain would make the inequality true. To show all is false, we need only supply one domain number where the inequality is false and then all is false.

The inequality is certainly true for some domain numbers, like, \(G(2.9) < 0\), since the point corresponding to \(2.9\) sits below the horizontal axis. However, the original statement said that all of the domain numbers would make the inequality true. They don’t all make the inequality true. We have provided a domain number where the inequality is false. We have provided a counterexample. The stament about all domain values is false.

\(\blacktriangleright \) \(G(x) < 10\) for all values of \(x\) in the domain.

The graphs makes this statement seem true. All of the points are positioned vertically below the height of \(10\). This makes us think that there is no counterexample.

Since we think the statement might be true, we could attempt to explain why we think it is true. But, we are interested in counterexamples here, so we will just move on.

\(\blacktriangleright \) \(G(x) > 4\) for each value of \(x\) in the domain such that \(x > 0\).

“for each” is just another way of saying “all”.

This statement is a claim about the function values, but not all of the function values. It refers to the function values on a subset of the domain. It makes a claim about all of the positive domain numbers. This is still an “all” statement. This “all” statement does not cover the whole domain. It covers a subset of the domain.

This statement is false. Our counterexample is \(1\).

First, \(1\) is a domain number included in the described domain subset, because \(1 > 0\).

Secondly, \(G(1) \approx 1.5 < 4\).

\(1\) is just one number in the domain subset. The original statement used the word all and that means that every positive number in the domain would make the inequality true. To show all is false, we need only supply one number in the domain subset where the inequality is false and then all is false.

The inequality is certainly true for some domain numbers, like, \(G(3.1) > 4\), since the point corresponding to \(3.1\) sits at a height greater than \(4\). However, the original statement said that all of the positive domain numbers would make the inequality true. They don’t all make the inequality true. We have provided a positive domain number where the inequality is false. We have provided a counterexample.

Notice that \(3\) is not a counterexample. \(3\) is a positive number, but the statement was not about positive numbers. It referred to positive domain numbers and \(3\) is not in the domain.

\(\blacktriangleright \) \(G(x) < 2\) for every value of \(x\) in the domain such that \(x < 0\).

The graphs makes this statement seem true. Not all of the points on the whole graph are positioned vertical below the height of \(2\), but that is not what the statement claims. The statement was about function values at negative domain numbers. For negative domain numbers, the corresponding points are below a height of \(2\). This makes us think that there is no counterexample.

Since we think the statement might be true, we could attempt to explain why we think it is true. But, we are interested in counterexamples here, so we will just move on.

\(\blacktriangleright \) Note: Just because YOU might not be able to find a counterexample does not mean the statement is true. Identifying a counterexample shows that the statement is false. Not finding a counterexample means you cannot draw any conclusions. Showing a statement is true requires more explanation than just you could not find a counterexample.