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How do you show that a conjecture is true—or false?

As you might imagine, not all conjectures turn out to be true! Technically, a conjecture is a statement that appears to be true, but has not yet been proved. In this lesson, you will learn some ways to prove or disprove a conjecture.

Let's start with the basics. If you think a conjecture is false, you might take a different approach to proving it is false than if you believed the conjecture to be true. Compare the two approaches on the tabs below.

False Conjecture

True Conjecture

To prove a conjecture is false, you might look for examples that show that the conjecture is false. In geometry, these are called counterexamples. If you can find just one counterexample, the conjecture is false.

frustrated student looking down at notebook and calculatorstudent working on a math problem on chalkboard

For example, suppose you observe that every Friday so far your math teacher has given the class a quiz. Based on this observation, you might state the conjecture that every Friday you will have a quiz in math. If you then observe that one Friday you do not have a quiz, your conjecture is false.

Day of Week Ending Letter
Sunday y
Monday y
Tuesday y
Wednesday y
Thursday y
Friday y
Saturday y
Proving that a conjecture is always true requires a little more work. To prove that something is always true, you must show that it is true for all possible situations. Just one example won't work.

For example, someone might make the conjecture that the names of all days in the week end with the letter "y." We can prove that conjecture by looking at every possible situation involving the name of a day in a week, as in the table to the left.

Since every possibility ends with a "y," the original conjecture is true.

Question

What is the main difference between proving a conjecture is false and proving a conjecture is true?

To prove a conjecture is false, you just need one example. However, to prove a conjecture is true, you need to look at all possible examples.