Download 1.3 Using Reasoning to Find a Counterexample

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
12 = 3 + 4 + 5
22 = 4 + 5 + 6 + 7
6=1+2+3
10 = 1 + 2 + 3 + 4
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:
1x8+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
1x8+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