s3_3 - 3 MATHEMATICAL INDUCTION 83 3 Mathematical Induction...

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3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P ( n ) be a predicate with domain of discourse (over) the natural numbers N = { 0 , 1 , 2 ,... } . If (1) P (0), and (2) P ( n ) P ( n + 1) then nP ( n ). Terminology: The hypothesis P (0) is called the basis step and the hypothesis, P ( n ) P ( n + 1), is called the induction (or inductive) step . Discussion The Principle of Mathematical Induction is an axiom of the system of natural numbers that may be used to prove a quantiﬁed statement of the form nP ( n ), where the universe of discourse is the set of natural numbers. The principle of induction has a number of equivalent forms and is based on the last of the four Peano Axioms we alluded to in Module 3.1 Introduction to Proofs . The axiom of induction states that if S is a set of natural numbers such that (i) 0 S and (ii) if n S , then n + 1 S , then S = N . This is a fairly complicated statement: Not only is it an “if . .., then . ..” statement, but its hypotheses also contains an “if . .., then . ..” statement (if n S , then n + 1 S ). When we apply the axiom to the truth set of a predicate P ( n ), we arrive at the ﬁrst principle of mathematical induction stated above. More generally, we may apply the principle of induction whenever the universe of discourse is a set of integers of the form { k,k + 1 ,k + 2 ,... } where k is some ﬁxed integer. In this case it would be stated as follows: Let P ( n ) be a predicate over { k,k + 1 ,k + 2 ,k + 3 ,... } , where k Z . If (1) P ( k ), and (2) P ( n ) P ( n + 1) then nP ( n ). In this context the “for all n ”, of course, means for all n k .

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3. MATHEMATICAL INDUCTION 84 Remark 3.1.1 . While the principle of induction is a very useful technique for proving propositions about the natural numbers, it isn’t always necessary. There were a number of examples of such statements in Module 3.2 Methods of Proof that were proved without the use of mathematical induction. Why does the principle of induction work? This is essentially the domino eﬀect. Assume you have shown the premises. In other words you know P (0) is true and you know that P ( n ) implies P ( n + 1) for any integer n 0. Since you know P (0) from the basis step and P (0) P (1) from the inductive step, we have P (1) (by modus ponens ). Since you now know P (1) and P (1) P (2) from the inductive step, you have P (2). Since you now know P (2) and P (2) P (3) from the inductive step, you have P (3). And so on ad inﬁnitum (or ad nauseum). 3.2. Using Mathematical Induction. Steps 1. Prove the basis step. 2. Prove the inductive step (a) Assume P ( n ) for arbitrary n in the universe. This is called the induction hypothesis . (b) Prove
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s3_3 - 3 MATHEMATICAL INDUCTION 83 3 Mathematical Induction...

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