L4 - MATH 135 Lectures IV/V Notes Winter 2009 Mathematical...

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MATH 135 Winter 2009 Lectures IV/V Notes Mathematical Induction The second technique of proof at which we’ll look is mathematical induction . This is a technique that is normally used to prove that a statement is true for all positive integers n . Motivation Why would we want to do this? Suppose that we had a sequence { x n } defined by x 1 = 1 and x r +1 = 2 x r + 1 for r 1. What is x 2 ? x 3 ? x 4 ? Do you see a pattern? (Beware of patterns!) What if we wanted to know x 2008 ? What would we have to do? It doesn’t make sense to calculate x 2 through to x 2007 individually, and it does look like we have a pattern. Is there a way to prove that x n = 2 n - 1 for all positive integers n ? This would allow us to be able to state the value of x 2008 or x 23487 or whatever. This is where induction comes in. Technical Details Induction is a technique for proving statements of the form P ( n ) where n P . (For example, P ( n ) = “ x n = 2 n - 1”.) This technique relies on Principle of Mathematical Induction (POMI) Let P ( n ) be a statement that depends on n P . If (i) P (1) is true, and (ii) P ( k ) is true P ( k + 1) is true, then P ( n ) is true for all n P . Why does this work? Suppose that we know (i) and (ii) to be true about P ( n ). (i) says P (1) is true (ii) says “If P (1) is true, then P (2) is true” so P (2) is true (ii) then says “If P (2) is true, then P (3) is true” so P (3) is true Following this along, P ( n ) is true for all n P A couple of analogies: dominos, robot climbing ladder Proofs by Induction There are three parts to a proof by induction: i) Base Case – Prove statement for smallest admissible value of n (usually n = 1) ii) Induction Hypothesis – Suppose that P ( k ) is true for some k P iii) Induction Conclusion – Prove that P ( k + 1) is true based on the hypothesis that P ( k ) is true
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Example A sequence { x n } is defined by x 1 = 1 and x r +1 = 2 x r + 1 for each positive integer r 1. Prove that x n = 2 n - 1 for every positive integer n .
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