Inductive Reasoning

Inductive Reasoning ? Making a rule after seeing several examples; whenever we arrive at a general statement by investigation.

Examples:

1)         The cafeteria has run out of burger buns everyday for a week.  Jim concludes the cafeteria does not have enough buns in supply

2)                  The pattern of the sequence 5, 10, 15 ? is 20, 25, etc.

Conjectures

Rules, general statements or conclusions based on inductive reasoning may or may not be true.

If they are not true, you can often show that the conclusion is false by finding a counter-example.

Examples:

            1)         The sum of any 2 prime numbers is an even number

                        Counter-example:         2+5 = 7

                        Therefore the statement is false.

2)                  For the graph y = x^x , x > 0

Counter-example:         x = -0.5            y = (-0.5)^-0.5     y =ö

                                    x = -1               y = (-1)^-1          y = -1

(If we graph y = x^x, we can conclude by looking at the graph that the statement is false)

Deductive Reasoning

This type of reasoning is where you use a rule to make a decision.

You start with a general statement which has been accepted as true & apply it to other cases.

Examples:

            1)         If you have a bus pass, you can get on the bus.  Jim has a bus pass.

                        Conclusion:       Jim can get on the bus

2)                  Find x

                        x² = 5² + 15²

                        x = 13

Example:

Prove the following problem by induction and deduction:

Choose a number.  Double it, Add 5, Add your original number, Add 7, Divide by 3, and Subtract your original number.  What is your conclusion?

Solved by induction:

            Choose:            3                      5                      0                      8

                                    6                      10                    0                      16

                                    11                    15                    5                      21

                                    14                    20                    5                      29

                                    21                    27                    12                    36

                                    7                      9                      4                      12

                                    4                      4                      4                      4

            Therefore the number is 4

Solved by deduction:

            Let ?x? be the number:

            2x

            2x+5

            3x+5

            3x+5+7

            x+4

            4

            Therefore the number is 4

Indirect Reasoning

Indirect Reasoning is reasoning by contradiction.

Here you assume that the desired conclusion is false if the assumption the leads to a contradiction then it can be concluded that the assumption was incorrect and therefore the conclusion is in fact true.