Download 1.2 Inductive Reasoning

1.2 Inductive Reasoning
Inductive Reasoning
If you were to see dark, towering clouds
approaching what would you do? Why?
Inductive Reasoning
• When you make a conclusion based on a
pattern of examples or past experience, you
are using inductive reasoning.
• Inductive reasoning: looking for patterns and
making observations
Conjecture
• Conjecture: an unproven statement that is
based on a pattern or observation
Ex: The sum of two odd number is _________.
1 + 1 =2
5 + 1 = 6 3 + 7= 10
3 + 13 = 16
21 + 9 = 30
Conjecture: The sum of any two odd numbers is
even.
Conjectures
• The difference of any two odd numbers is
_______.
The difference of any two odd numbers is even.
Stages of Inductive Reasoning
1. Look for a pattern. Look at several examples.
2. Make a conjecture. Use examples to make a
general conjecture. Change it if needed.
3. Verify the conjecture. Use logical reasoning
to verify that the conjecture is true in all
cases.
Counterexamples
• A conjecture is an educated guess.
– Sometimes it is true and other times it is false
– True for some cases does not prove true in
general
– To prove true, have to prove true in all cases
– Considered false if not always true.
– To prove false, need only 1 counterexample
• Counterexample: an example that shows a
conjecture is false.
Counterexamples
Conjecture: If the product of two numbers is
even, the numbers must be even.
Counterexample: 7 × 4 =28
28 is even but 7 is not
Counterexamples
Conjecture: Any number divisible by 2 is
divisible by 4.
Counterexample:
6 is divisible by 2, but not by 4.
Review
• What is a conjecture?
– An unproven statement that is based on a pattern
or observation
• How can you prove a conjecture is false?
– By finding just one counterexample.