lec0224 - Well Ordering Principle Every non-empty subset of...

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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 be used to prove anything that we can prove inductively. First, I’ll show you another proof of why mathematical induction works, using the well-ordering principle: Let an open statement satisfy the requirements I showed you the other day for proving an open statement by induction, so we have s(1) s(k) s(k+1), for all positive integers k. Now, let’s assume that these two conditions could hold while the statement s(n) is NOT true for all positive integers n. This means that we can create a set F = { t Z + | s(t) is false } that is non-empty. Since this set is a subset of Z + , it must contain a smallest element. Let that smallest element be w. (Thus s(w) is NOT true is our assumption.) We know that w 1 since that is the first requirement for an inductive proof to hold. Thus, we know that w > 1, which means that w-1 Z + . But, we know that if w-1 Z + , AND that s(w-1) is TRUE because we assumed that w was the SMALLEST value for which s(n) was NOT true. Combining that with the fact that s(k) s(k+1), for all positive integers k, then we can plug in k = w-1 to get: s(w-1) s((w-1)+1) or s(w-1) s(w), implying that s(w) is true. But, this contradicts our assumption that s(w) is false.
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This document was uploaded on 07/14/2011.

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lec0224 - Well Ordering Principle Every non-empty subset of...

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