Math_Strong_Induction

Mathematical Induction and Strong Induction

Scope:

1)Mathematical Induction

2)Strong Induction

1. Mathematical Induction

1.1. Introduction & Definition

Many mathematical statements assert that a property is true of all positive

integers.

Example: U=Z+,

Show that

Mathematical induction is a proof technique that is used to prove a property of a set of positive integers, which is based on the rule of inference:

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Two Steps:

1.Basis step: Show that

2.Inductive step: Show that

Example: We have a line of people.

We know that the first person will be told the secret and a person

will communicate the secret to the next person in the line. Show

that all people will know the secret.

1.Basis step: Since we know that the first person will be told the

secret, it follows

2.Inductive step: Let be arbitrary.

------------------------------------------------- Hypothesis

person will communicate the secret to the next person+1 in the

line ------------------------------------------------- Given fact

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Because is arbitrary, by universal

generalization. Therefore,

1.2. Examples

General example:

To prove P(n) is true for all integers n ≥ c. We do the following steps:

1. Basis step: Show that P(c) is true (where c is the smallest element).

2. Inductive step: Show that P(k) → P(k + 1) is true for all k ≥ c, i.e. show that

P(k) → P(k + 1). (Assume P(k) is true and show that P(k +1) is true.) A proof by mathematical induction follows a fixed pattern.

1.3. Exercise

Suppose that we have an infinite ladder and we know two things:

1.We can reach the first step of the ladder.

2.If we can reach a particular step of the ladder, then we can reach the next step.

Show that we can reach every step on this ladder.

2. Strong Induction (also called generalized induction)

2.1. Introduction & Definition

Introductory Example:

Suppose that we have an infinite ladder and we know two things:

1.We can reach the first and second steps of the ladder.

2.If we can reach a particular step of the ladder, then we can reach two steps higher.

Show that we can reach every step on this ladder.

Rule of inference:

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Two Steps:

1.Basis step: Show

2.Inductive step: Show

or for an

arbitrary k

Solution to the introductory example:

1.Basis step: We know that we can reach first step →

2.Inductive step: for an arbitrary k, we prove two cases:

Case#1:

We know that we can reach the second step

Case#2:

Assumption

Given fact

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2.2. Example: Fundamental Theorem of Arithmetic