COT3100Induction01 - Well Ordering Principle Every...

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Well – Ordering Principle Every non-empty subset of Z + (the positive integers) contains a smallest element. Essentially the set Z + is well-ordered. Well, DUH!!! This sounds like a really useless principle. But it can actually be used to prove a wide range of statements. Furthermore, it can be used to show that mathematical induction is a valid proof technique. First I will show you an example of a proof that utilizes the well ordering principle, then I will show how the Well- Ordering Principle implies mathematical induction. Consider proving the following summation to be true for all positive integers n: = + = n i n n i 1 2 ) 1 ( We will prove this statement by contradiction. Assume that this statement is false for at least one positive integral value of n. Let S be the set of all positive integers n for which the statement above is false. Using our assumption, S is non-empty. By the Well-Ordering Principle, S must contain a smallest element. Let this smallest element be k. We know that k can NOT be 1 because the statement above is true for n=1. (To verify, check that the LHS = 1 and the RHS = 1(2)/2 = 1.)
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Thus, k > 1. Furthermore, it follows that k-1>0. Since k-1 is a positive integer less than k, we can deduce that the formula is true for k-1. This means that - = + - - = 1 1 2 ) 1 ) 1 )(( 1 ( k i k k i Now, consider computing the following sum: = k i i 1 k i i k i k i + = - = = 1 1 1 k k k + - = 2 ) 1 ( 2 2 ) 1 ( k k k + - = 2 ) 2 1 ( + - = k k 2 ) 1 ( + = k k But, this contradicts our assumption that k was the smallest positive integer for which the formula was false. Thus, our assumption, that such an integer exists is false, and the formula must be true for all positive integral values of n.
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Here is the basic idea behind mathematical induction: The goal of mathematical induction is to prove an open statement s(n) for all non-negative integers n, or all positive integers n. We can do this in the following manner: 1) Show that s(1), the base case, is true. 2) Show that s(k)
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This document was uploaded on 07/14/2011.

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COT3100Induction01 - Well Ordering Principle Every...

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