What statements do you consider matters of fact, beyond the need for proof? What about this one?

You might think this is a true statement that everyone agrees upon. If you look out your window right now, there's a good chance that you'll see a blue sky. However, a sentence like The sky is blue should make you think of conjectures, statements that appear to be true but have not been proven.

The simplest way to show that a conjecture is not true is to provide a counterexample. In this case, you would just need to show a picture like this one to disprove the conjecture that the sky is blue. At night the sky is black, and if you ever take the time to watch a sunrise or sunset you'll see many more colors. And, of course, when it's raining, the sky is usually gray.

You'll encounter many conjectures in your life--ones that you hear and ones that you, yourself, think or say. To understand and evaluate a situation described by a conjecture, you'll need to gather evidence, including several examples.

You may not be sure which way the evidence points--for or against the conjecture. In these cases, you may want to organize your findings visually, using lists or some kind of diagram. This lesson will show you how to use a particularly useful diagram for proving or disproving a conjecture.