- Inductive Reasoning is the process of arriving at a general conclusion from looking at specific examples
- The sum of two even numbers is even:
2 + 4 = 6
4 + 8 = 12
6 + 10 = 16
- The first step is to form a hypothesis or conjecture
- This is just a statement of what we believe to be true.
- In the above example The sum of two even numbers is even
- The problem with inductive reasoning is that it might not be correct.
- Example
The sum of an two single digit numbers is a single digit number
1 + 3 = 4
2 + 5 = 7
3 + 6 = 9
6 + 7 = 13
- The last case, 6 + 7 = 13, is a counterexample or an example where the conjecture does not hold true.
- It only takes one counter example to prove the hypothesis false.
- The problem with inductive reasoning is that it does not guarentee that the hypothesis is correct.
- But it is something we use in everyday life (often). Or at least we should.
- We use inductive reasoning anytime we are looking for a pattern
- Example
What number comes next? 37 32 27 22 17
Form a hypothesis -
subtract five from the previous number to find the next number.
Use the hypothesis to find the next number
17 - 5 = 12
- Deductive reasoning is the process of proving a specific conclusion from one or more general statements.
- This is the basic tool of Mathematicians.
- If the general statements are true, then the conclusion reached by deductive reasoning is also true.
- Example
Pick a number
Multiply it by 6
Subtract 12
Divide by 3
Add 4
Divide by 2
This will give you your original number.
(4) 4 * 6 = 24 (1) 1 * 6 = 6
24 - 12 = 12 6 - 12 = -6
12 / 3 = 4 -6/3 = -2
4 + 4 = 8 -2 + 4 = 2
8 / 2 = 4 2/2 = 1
Or let x be the starting number
(x) x*6
x*6-12
(6x-12)/3 = 2x-4
2x-4+4 = 2x
2x/2 = x
- In the latter case, we have proven, by deduction, that the hypothesis is true.