B. Validity of Conjectures and Counterexamples

So far you have used inductive reasoning to make conjectures. A conjecture is said to be true if it gives a prediction that is always correct. You have probably noticed that many of the conjectures from the first part of this lesson will not always give a correct prediction, and so are false. Typically, it is easier to show that a conjecture is false than it is to show that it is true.

Consider the following conjecture: All Frenchmen live in France.

If you can show that a single Frenchman does not live in France, you have shown the conjecture is not true. An example that shows a conjecture is not always true is called a counterexample. Only one counterexample is required to show that a conjecture is false.

Example 1

The following table shows some conjectures and a counterexample that shows each is false.