# 2.6: Conjectures and Counterexamples

- Page ID
- 2141

Educated guesses and examples that disprove them.

A **conjecture** is an “educated guess” that is based on examples in a pattern. A **counterexample** is an example that disproves a conjecture.

Suppose you were given a mathematical pattern like \(h = \dfrac{−16}{t^2}\). What if you wanted to make an educated guess, or conjecture, about \(h\)?

Use the following information for Examples 1 and 2:

A car salesman sold 5 used cars to five different couples. He noticed that each couple was under 30 years old. The following day, he sold a new, luxury car to a couple in their 60’s. The salesman determined that only younger couples by used cars.

Example \(\PageIndex{1}\)

Is the salesman’s conjecture logical? Why or why not?

**Solution**

It is logical based on his experiences, but is not true.

Example \(\PageIndex{2}\)

Can you think of a counterexample?

**Solution**

A counterexample would be a couple that is 30 years old or older buying a used car.

Example \(\PageIndex{3}\)

Here’s an algebraic equation and a table of values for \(n\) and \(t\).

\(t=(n−1)(n−2)(n−3)\)

\(n\) | \((n−1)(n−2)(n−3)\) | \(t\) |
---|---|---|

1 | \((0)(−1)(−2)\) | 0 |

2 | \((1)(0)(−1)\) | 0 |

3 | \((2)(1)(0)\) | 0 |

**Solution**

After looking at the table, Pablo makes this conjecture:

The value of \((n−1)(n−2)(n−3)\) is 0 for any number n.

Is this a true conjecture?

This is not a valid conjecture. If Pablo were to continue the table to n=4, he would have see that \((n−1)(n−2)(n−3)=(4−1)(4−2)(4−3)=(3)(2)(1)=6\)

In this example \(n=4\) is the counterexample.

Example \(\PageIndex{4}\)

Arthur is making figures for an art project. He drew polygons and some of their diagonals.

From these examples, Arthur made this conjecture:

If a convex polygon has \(n\) sides, then there are \(n−2\) triangles formed when diagonals are drawn from any vertex of the polygon.

Is Arthur’s conjecture correct? Or, can you find a counterexample?

**Solution**

The conjecture appears to be correct. If Arthur draws other polygons, in every case he will be able to draw \(n−2\) triangles if the polygon has n sides.

*Notice that we have***not proved** *Arthur’s conjecture, but only found several examples that hold true. So, at this point, we say that the conjecture is true.*

Example \(\PageIndex{5}\)

Give a counterexample to this statement: Every prime number is an odd number.

**Solution**

The only counterexample is the number 2: an even number (not odd) that is prime.

## Review

Give a counterexample for each of the following statements.

- If \(n\) is a whole number, then \(n^2 >n\).
- All numbers that end in 1 are prime numbers.
- All positive fractions are between 0 and 1.
- Any three points that are coplanar are also collinear.
- All girls like ice cream.
- All high school students are in choir.
- For any angle there exists a complementary angle.
- All teenagers can drive.
- If \(n\) is an integer, then \(n>0\).
- All equations have integer solutions.

## Review (Answers)

To see the Review answers, open this PDF file and look for section 2.5.

## Resources

## Vocabulary

Term | Definition |
---|---|

conjecture |
A conjecture is an educated guess that is based on examples in a pattern. |

counterexample |
A counterexample is an example that disproves a conjecture. |

fraction |
A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a .rational number |

## Additional Resources

Interactive Element

Video: Inductive Reasoning

Activities: Conjectures and Counterexamples Discussion Questions

Study Aids: Types of Reasoning Study Guide

Practice: Conjectures and Counterexamples

Real World: Conjectures And Counterexamples