Three lectures in to the Principle of Simple Induction, and I've learned a few things.
First, there seems to be no ideal medium for expressing mathematical ideas live to a large group of people. Slides are too canned --- they give the harmful impression that mathematics is completely clean and slick. Blackboards are volatile (your work is gone next lecture), and often suffer from poor lighting and the obscuring effects of chalk dust. Typing live in a mathematical editor such as Lyx doesn't make drawings or diagrams easy, and it still makes the math look too slick. And my latest toy, a tablet, is a bit strange to write legibly on, and I live in fear of losing my stylus.
Second, examples that I begin by thinking will illustrate one feature sometimes turn out to illustrate another.
For example, my gut reaction to the claim that for every natural number n, 3 raised to the power of n will end with unit digit either 1, 3, 7, or 9, was that here is a case that will likely require multiple base cases. As I began to sketch a proof in my head, I realized that all that's required is the case for n=0.
Similarly, some colleagues had once assigned the question "How many odd subsets does a set with n members have?" as a typical exercise in induction. I brainstormed this with one section, and the conjecture that there were (2^n)/2 odd subsets emerged. The case where n=0 seems like a special case, since the empty set has an even subset (itself) and no odd subsets, whereas for other values of n there seem to be the same number of even and odd subsets.
One student firmly defended her position that no special treatment is required for n=0 --- the induction step only needs to use that fact that there is a 1-1 correspondence between the even subsets of S\{x} and the odd subsets of S, and a 1-1 correspondence between the odd subsets of S\{x} and the odd subsets of S. I was very skeptical at first, and then I realized that her argument meant that no induction was needed --- the set of odd subsets of a set with n members can be put in 1-1 correspondence with the set of all the subsets (odd and even) of a set of n-1 members.
So, although induction would work as a proof technique, it was not required.
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2 comments:
I read your post first without realizing that it was the work of the instructor... and I seriously started worrying about my level being not up to par.
..phew!
mt
I'm finding that it's true, there is no ideal medium for presenting mathematical proof or statement. The chalk board is one that I detest the most. One TA I used abused the erasing factor of the blackboard when working with matrices. Instead of writing a new line he just erased what he had and wrote over it. That was a problem for someone like me who was copying it down. The tablet, though the writing's legibility decreased a little bit, not enough for it to be a problem.
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