13 - LECTURE 13 INDUCTION 6.1 Courage is an accumulation...

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Unformatted text preview: LECTURE 13: INDUCTION ( § 6.1) Courage is an accumulation of small steps. George Konrad 1. Another Axiom Definition 1. If S ⊆ N is a subset of the natural numbers, and S 6 = ∅ , then S has a minimal element. This fact is called the well-ordering property for the natural numbers. We take this fact as an axiom. A set is well-ordered if every subset has a minimal element. Example 2. Is Q well-ordered? How about [0 , 1]? One way to picture the well-ordering property is label the steps of an infinite staircase by the natural numbers. For each number in the set S , put an x on that step. Intuitively there must be a smallest step with an x on it. How does one climb an infinite staircase? Mathematicians break it into two steps. Base case: Climb the first stair. Inductive case: Once you’ve climbed the first k stairs, climb the ( k + 1)st stair. This climbing procedure is called Proof By Induction . How do we know we’ve climbed every stair?...
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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13 - LECTURE 13 INDUCTION 6.1 Courage is an accumulation...

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