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# lec6 - Math 117 6 Natural and real numbers The set of...

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Math 117 – April 14, 2011 6 Natural and real numbers The set of natural numbers is N = { 1 , 2 , 3 , 4 , . . . } . We will assume that the operations of addition (+) and multiplication ( · ), as well as the relation “ ” are well defined on the set of real numbers, and hence on the set of natural numbers as well. We further assume that N satisfies the following property. Axiom 6.1 (Well-ordering property) . If S N is a nonempty subset, then there exists an element m S , such that m k for all k S . The principle of mathematical induction is a consequence of the well-ordering property, and is a useful tool when proving theorems about natural numbers. Theorem 6.2 (Principle of mathematical induction) . Let P ( n ) be a statement for every n N . Then P ( n ) is true for all n N , provided ( a ) P (1) is true (basis for the induction) ( b ) k N , if P ( k ) is true, then P ( k + 1) is true (inductive step). A generalization of the math induction is the principle of strong induction.
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