1 2 ABSTRACT ALGEBRA An equivalent statement, the Well-Ordering Principle for N, will be formulated later in the text. Another equivalent statement is the following. The Principle of Complete Mathematical Induction. Suppose that {P(0), P(l),...} is a set of mathematical statements indexed by the natural numbers. Suppose further that the following two statements are true: (a) P(0) is true (b) For any natural number n, the truth of P(k) for all k less than n implies the truth of P(n). Then all of the statements P(0), P(l),.. . are true. You will see some examples of proofs by (complete) mathematical induction in the text, and you will have some opportunities in the exercises to practice creating such proofs yourself.

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