goldch1 - Contents I Induction I.A I.B I.C The Planet X...

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Contents I Induction 1 I.A The Planet X Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 I.B Induction Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I.C Examples for Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
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I Induction I.1 I Induction Induction is a mathematical technique for proving an infinite sequence of statements S ( n ) indexed by integers. To explain it informally first, we begin with an excursion. I.A The Planet X Rule Assume there is some planet X out there, similar to our planet but with one significant diFerence. The weather on planet X is determined by the following rule: “If it rains one day, it also rains the next day.” Now you land on planet X and A: It does not rain the day you arrive. What can you conclude? (a) It will never rain on planet X . (b) It has never rained on planet X . (c) It did not rain yesterday on planet X . (d) It will rain tomorrow on planet X . B: It rains the day you arrive. What can you conclude? (e) It rains every day on planet X . (f) It will rain tomorrow on planet X . (g) It rained yesterday on planet X . (h) It will rain every day from now on. Please think about each statement and find all those statements that are warranted by the rule. Mathematically, the weather rule for planet X can be stated as follows: ±or every day n , we have a statement — that may be true or false: S ( n ): It rains on the day numbered n . We also have a rule : S ( n ) = S ( n + 1), (read: S ( n ) implies S ( n + 1)). In plain English, if it rains one day (with number n ), it will rain the next day (with number n + 1). Assume you arrive on day 10. Alternative (A) says that S (10) is false , as it does not rain on that day. From a false hypothesis, everything can happen! Thus we do not know whether it will rain on day 11, day 110, day 110000. .. But we can conclude that it didn’t rain yesterday! Had it rained on day 9, S (9) were true and then also S (10) must be true, as S (9) entrains S (10) by the rule given. Going back in time, we obtain that it has never rained before you arrived.
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I.2 I.B Induction Principle Alternative (B) says that S (10) is true — it rains the day you arrive. Thus, for n = 10, the hypothesis is true and so must be S (11) — it will rain tomorrow. Going forward, we see that S (10 + n ) must be true for every n 0, i.e. it will rain every day to come! As to yesterday or any other day in the past, we cannot conclude anything! In conclusion, (b), (c), (f), (h) are true statements that can be inferred from the given rule, but we have no way of knowing whether (a), (d), (e) or (g) are true, they cannot be concluded or validated from the rule. Mathematical Induction works precisely this way.
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goldch1 - Contents I Induction I.A I.B I.C The Planet X...

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