12.2 Mathematical Induction

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Figure 12.6 An inductive proof.

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Induction proceeds from observation to conjectures to proof. Conjectures that are proved become laws. For example, we may observe that 1 = I 2 , 1 + 3 = 2 2 , 1 +- 3 + 5 = 32 and 1 + 3 + 5 + 7 = 4 2 . We recognize a pattern and so make the conjecture that

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1 + 3+ ... +(2m = (m + 1)

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We test the conjecture for the case ra = 3 and possibly others and then we prove the conjecture. Figure 12.6 shows a diagrammatic proof. Note that, at each stage, the square of size m is increased in size by adding 2m + 1 dots, as indicated by the boxes1. Typically, inductive reasoning is not so straightforward. More often than not, the conjectures we make are unfounded. They do not stand up to proof and have to be discarded or, at best, modified in some way. In order to improve the effectiveness of inductive reasoning, it is important to limit the amount of guesswork, reducing induction as far as possible to deduction. Of course, it is never possible to eliminate guesswork altogether otherwise the creative element of inductive reasoning would be eliminated, and that is too much to expect.

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The Principle of Mathematical Induction

The principle of mathematical induction provides a method of proving that a property P predicated on natural numbers is true for all natural numbers2. An example is the predicate S defined by S.n = < I k : l < k ^ n : k > = An(n+l)

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(12.6)

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(Using the dotdotdot notation: S.n = l + 2 + . . . + n = | n ( n + l ) . We prefer to use the quantifier notation in order to be precise and unambiguous. In particular, the quantifier notation makes it clear that S.O is well defined, whereas the dotdotdot notation appears to exclude the case that n equals 0.)

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The leftmost square in the diagram has zero dots, but you cannot see them! The inability to handle important special cases here, the case m = 0 is a major drawback of diagrams. 2 Recall that a natural number is a non-negative integer. So the natural numbers are the numbers 0, 1, 2, etc.

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1 2: Inductive Proofs and Constructions

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The essence of the principle of mathematical induction is that an arbitrary property P of the natural numbers is provably true for all natural numbers, n, if it is possible to prove (i) P.O is true, and (ii) for all n, P.(n+l) follows from the assumption that P.n is true. Example 12.7. To illustrate the principle let us apply it to the predicate S defined in (12.6). We begin by proving S.O. This first step is called the basis of the proof. We have

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0 =0

definition } empty-range rule to simplify the summation, arithmetic for the right side of the equality }

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{ true .

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reflexivity of equality }

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The next step, called the induction step, is to show that S.(n+l) follows from the assumption S.n. We have { { { definition } range splitting applied to the summation } assume S.n . That is, assume that

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( Z k : l^k^n+l : k ) = f ( n + l ) ( ( n + l ) + l)

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arithmetic }

The crucial step in this calculation is the bulleted step in which the assumption is made that S.n is true. This assumption is called the induction hypothesis. The final step is to cite the principle of mathematical induction to combine the basis and the induction step in the conclusion that property S.n is true for all natural numbers n. D