Name: ______________________ Date: ________________

Foundations of Mathematics 11

Chapter 1- Inductive and Deductive Reasoning 1.3 Using Reasoning to Find a Counterexample

Today’s Goal: To find examples that contradict given conjectures.

Let’s define Counterexample: In order to disprove or make a conjecture false we must find one example of the statement that does not

satisfy the conjecture. Conjecture: Natural numbers can be written as the sum of consecutive numbers.

Here are some examples:

9 = 4 + 5 6 = 1 + 2 + 3

12 = 3 + 4 + 5 10 = 1 + 2 + 3 + 4

22 = 4 + 5 + 6 + 7 29 = 14 + 15

Find a counterexample to this conjecture. How can you be sure that your counterexample is correct?

Example: Matt found a interesting numeric pattern:

1 x 8 + 1 = 9

12 x 8 + 2 = 98

123 x 8 + 3 = 987

1234 x 8 + 4 = 9876

Matt thinks that this pattern will continue. Search for a counterexample to Matt’s conjecture.

A B

1 1 x 8 + 1 9

2 12 x 8 + 2 98

3 123 x 8 + 3 987

4 1234 x 8 + 4 9876

5

6

7

8

9

10

The pattern holds true until the ______________digit is added we have a counterexample

Several conjectures are given. Decide whether each conjecture is true or false. If it is true,

write to explain why. If it is false, give a counterexample.

a. A number that is not positive is negative.

b. If 1 is added to an odd number, the result is always an even number.

c. The square of a number is always greater than the number

d. If two angles are acute, their sum is less than 180°.

e. Every rectangle is a square.

f. Every square is a rectangle.

In Summary

Key Idea:

Once you have found a counterexample to a conjecture, you have

______________________ the conjecture. This means that the conjecture

is _______________________.

You may be able to use a counterexample to help you ________________

a conjecture.

Need to Know

_______________________ is enough to disprove a conjecture.

Even if you cannot find a counterexample, you ______________________

__________________________________________________________ .

Any supporting evidence you develop while searching for a

Counterexample, however, does __________________________________

Assignment: p. 22 # 1 – 8, 10, 17, 19