Inductive Reasoning

Goals:

Define inductive and deductive reasoning
Recognize the value of inductive reasoning
Recognize the the potential pitfalls of inductive reasoning
Apply inductive reasoning to solve problems.

Inductive Reasoning

Inductive Reasoning is the process of reasoning to a general conclusion through observations of specific cases
Scientists often use inductive reasoning (The Scientific Method)
This is pattern recognition.
What comes next?: 2 4 6 8 10 12
What comes next?: 1 1 2 3 5 8 13

Can we say anything about the sum of two even number?


2 + 4 =  6    4 + 4 =  8     6 +  6 = 12
2 + 6 =  8    4 + 6 = 10     6 +  8 = 14
2 + 8 = 10    4 + 8 = 12     6 + 10 = 16

It looks like the sum of two even numbers is even. This is inductive reasoning.

Inductive reasoning involves
- Look for a pattern
- Use the pattern to predict what will happen
What happens when we add an odd and even number?
This type of reasoning is often correct, but does not have to be.
Prime numbers are numbers only divisible by themselves and 1
- 2 is prime
- 3 is prime
- 5 is prime
- 7 is prime
- We might conjecture, (or guess) that all odd numbers are prime
- Can we conclude that 9 is prime?
A counter example is an example that matches the pattern but is not right.
This is the main problem with inductive reasoning, we don't guarantee that there is not a counter example.
This often occurs when you observe insufficient number of cases
Or when you see a false pattern.

Deductive Reasoning

Deductive Reasoning is the process of reasoning to a specific conclusion from a general statement.
Mathematicians use deductive reasoning.
(There is a special case of this called Induction, but don't worry about that ...)

Statement: The sum of any two even integers is even.

Let m and n be two even integers.
    m = 2 * a and n = 2 * b  (This is the definition of even numbers)
    m + n = 2*a + 2 * b  
          = 2 * (a +  b) Distributive property
   Let  y = a+b 
    m + n = 2*y
   and 2*y  is an even number.

Examples

Look at the following problems as time allows. Make sure that you write down the conjecture for each.

12, page 5
14, page 5
18, page 5
22, page 6
32, page 6
36, page 7
44, page 7
46, page 7

Homework

Please do the following.

1-10 all.
11-35 odds
43-47 odds