8.4 Mathematical Induction

A Form of Proof

How do you climb infinite stairs?

– Start at the base of the staircase.– Step up on the first step.– Then, step up to the second step.– Next, step up to the third step..

.

.

.– Repeat:

From current position move up one step

Can we use this idea as a method of proof?

• First, show P(x) is true for x=0

– This is the base of the stairs

• Then, show that if it’s true for some value n, then it is true for n+1

– Show: P(n) P(n+1)

– This is climbing the stairs

– Let n=0. Since it’s true for P(0) (base case), it’s true for n=1

– Let n=1. Since it’s true for P(1) (previous bullet), it’s true for n=2

– Let n=2. Since it’s true for P(2) (previous bullet), it’s true for n=3

– Let n=3 …

– And onwards to infinity

• Thus, we have shown it to be true for all non-negative numbers

What is induction?

• An indirect method of proof• Involves the derivation of a general rule from one or

more particular cases

• The principle of mathematical induction is similar to the concept of domino stacking

Steps

1. The first expression must be true

(The first domino falls)

2. Assuming that the general expression is true

(Assume that some domino in the series falls)

3. Prove that the next expression is true

(Prove that the next domino in the series also falls)

4. If all of these events happen then we know by induction that all of the expressions are true and thus the original formula is true

(All the dominoes will fall)

Example 1

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Example 2

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