induction

# induction - Induction is a powerful method of proof often...

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Unformatted text preview: Induction is a powerful method of proof, often useful in cases where we wish to prove that some statement holds for all values of an integer parameter, n , starting at some smallest value n . For example, suppose we wish to prove that ∑ n i =1 i = n ( n + 1) / 2, for all n ≥ 1. Then n will be 1. The idea behind induction is to show two things: (i) if the statement ever “gets going” by being true for some n then it “keeps going” by staying true for all larger n (what we might acronymize as SIS (StartsImpliesStays), or ForeverAfter – if it ever Starts to be true then it Stays true forever after); and (ii) that the statement does in fact hold for the smallest value, n (the so-called base case). We usually do the “start” (or base) case first, and then show the “never stops” part (the so-called inductive step); but this is merely a convention, and as long as we have done both parts then we have shown that the state- ment holds for all n ≥ n . Another fanciful way to think of it is in terms of an inheritance law, eg, a fictitious law passed in 1800 saying that the children of every rich person must inherit all their parents’ wealth. This is not enough to guarantee that anyone at all is rich today, because perhaps no one happened to be rich in the first place. But if we also know that person A was rich (say in 1890) then it follows all the descendants of A are rich as well, including future descendants not yet born. The key fact about A (rich) then corresponds to the base case, and the inheritance law is the inductive step. Thus richness – once it starts – never stops (at least in this imaginary world). But it is a bit astonishing that one can ever show that a *mathematical* version of inheritance never stops – since we cannot expect Congress to pass laws guaranteeing our mathematics to come out right; and even if they did, we would not trust that kind of law to be a reliable guide to what is true. One of my favorite analogies is that of a contagious condition. Certain mathematical properties are behave like a contagious condition, in the sense that if one integer n “catches” that condition (has that property) then it “spreads” to the next one, n+1. Another highly suggestive one is a fire that spreads along a line of trees, from ones already ablaze to new trees further down the line. So how can we be sure that a math property is contagious, ie, spreads to the next n just from its being true for some particular? Wouldn’t we have to check it for the next n ? Yes and no. We do check it, but (when the method works) we check it by showing that there is a very nice connection between consecutive cases of n . That is, we show that there is a mathematical inheritance law for the statement, a law (or inductive step) showing (when it works) that the truth of the statement for a given value of n , say n = k 1 implies its truth for n = k + 1. This still seems astonishing; and there are two points to be aware of here: (i) induction does not always work, and (ii)...
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induction - Induction is a powerful method of proof often...

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