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2: Reasoning and Proof
2.6: Conjectures and Counterexamples

2.6: Conjectures and Counterexamples

Last updated

Nov 28, 2020
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- 2.5: Inductive Reasoning from Patterns
- 2.7: Deductive Reasoning

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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.

f-d_1a8e170aaf8620eaf6b280adec82dced4a1087956f5ea32a28204c7f+IMAGE_TINY+IMAGE_TINY.png — Figure $\PageIndex{1}$

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

Review

Review (Answers)

Resources

Vocabulary

Additional Resources

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