2.6: Conjectures and Counterexamples
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- Nov 28, 2020
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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
