12_Transcript

12_Transcript - 1 Induction 1.1 Welcome Welcome to the...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Induction 1.1 Welcome Welcome to the twelfth module in our course on proofs. Before going on, please do the assigned readings. 1.2 Introduction Induction is a common and powerful technique and should be your first choice whenever you en- counter a statement of the form for every integer n ≥ 1, P ( n ) is true where P ( n ) is a statement that depends on n . Here are two examples of propositions in this form. Proposition 1. For every integer n ≥ 1 n X i =1 i 2 = n ( n + 1)(2 n + 1) 6 . Often the clause “For every integer n ≥ 1” is implied and does not actually appear in the proposition, as in the following version of the same theorem. Proposition 2. The sum of the first n perfect squares is n ( n +1)(2 n +1) 6 . The second example uses sets, not equations. Proposition 3. Every set of size n has exactly 2 n subsets. 1.3 How To Use Simple Induction Our technique relies on the Principle of Mathematical Induction (POMI) sometimes called Simple Induction . We will see variations later on. Principle of Mathematical Induction (POMI) Let P ( n ) be a statement that depends on n ∈ P . If 1. P (1) is true, and 2. P ( k ) is true implies P ( k + 1) is true then P ( n ) is true for all n ∈ P . There are three parts in a proof by induction. Base Case Verify that P (1) is true . This is usually easy. In condensed proofs you will often see the statement “It is easy to see that the statement is true for n = 1.” It is best to write this step out completely. 1 Inductive Hypothesis Assume that P ( k ) is true, k ≥ 1. It is best to write out the statement P ( k ). Inductive Conclusion Using the assumption that P ( k ) is true, show that P ( k +1) is true . Again, it is usually best to write out the statement P ( k + 1) before trying to prove it. 1.4 Why Does Induction Work? The basic idea is simple. We show that P (1) is true. We then use P (1) to show that P (2) is true. And then we use P (2) to show that P (3) is true and continue indefinitely. That is P (1) ⇒ P (2) ⇒ P (3) ⇒ ... ⇒ P ( i ) ⇒ P ( i + 1) ⇒ ... 1.5 Two Examples of Simple Induction Our first example is very typical and uses an equation containing the integer n , and we have already seen it. Proposition 4. n X i =1 i 2 = n ( n + 1)(2 n + 1) 6 . Proof. We begin by formally writing out our inductive statement P ( n ) : n X i =1 i 2 = n ( n + 1)(2 n + 1) 6 . We must establish our Base Case. Base Case We verify that P (1) is true where P (1) is the statement P (1) : 1 X i =1 i 2 = 1(1 + 1)(2 × 1 + 1) 6 . As in most base cases involving equations, we can make our way from the left-hand side of the equation to the right side of the equation with just a little algebra. 1 X i =1 i 2 = 1 2 = 1 = 1(1 + 1)(2 × 1 + 1) 6 ....
View Full Document

This note was uploaded on 10/13/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.

Page1 / 9

12_Transcript - 1 Induction 1.1 Welcome Welcome to the...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online