Post #39,735
5/23/02 10:07:52 PM
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Questions for Ben Tilly.
Only two, and they're easy.
1) Somehow I got the impression that you were from Canada. This may date back to the InfoWorld Forum days, but I had the impression that you either lived there once, or were from there originally. Could you please clarify that for me?
2) The first grad class I had was in Real Analysis. For some reason I happened to recall that my first homework assignment was to prove that the cardinality of (0,1) was the same as [0,1]. This was before we had much to go on (no theorems about adding finitely many elements to an uncountable set, no definitions much to speak of). IIRC, I wrote a 2 page proof of this. I've thought off-and-on about this and I think it is a good puzzle. Construct such a proof with as few definitions as you can. How short can you make it? I think I still have the text we used and I can provide details about what we were able to use at the time the assignment was made if you wish. But I'm trying to think of the most eloquent proof I can muster using the fewest definitions I can. I'd be very interested to see what you thought - or how you would approach it anyway. And no, I will not accept the following form:
1) Assume x,y,z, etc.
2) Use the Contra-positive.
If you don't want to do this, fine. But I'm going to, and will probably publish my thoughts here for all to mock ;-)
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Post #39,751
5/23/02 11:43:32 PM
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Answers
1) I was born in the US, but grew up in Canada. I met my wife in Canada, who was in the same situation, and we both took out Canadian citizenship after the US State Department changed its rules to say that it was no longer pursuing dual citizenship cases. I currently reside in the US and am a dual citizen.
2) How much can you assume? If you can assume that cardinalities are strictly ordered, then (0,1) is a subset of [0,1], so its cardinality is less than or equal. However (0,1) has the same cardinality as (-0.5,1.5) as can be seen from the bijection x->2x-0.5. And [0,1] has cardinality less than or equal to (-0.5,1.5) because it is a subset. If either inequality was strict, then we would conclude that (0,1) has a cardinality less than itself, so they must be equal.
However given that I know enough about set theory to know the size of the sledgehammer I pulled out, the answer they were probably looking for was to construct the necessary bijection between the two sets. Many such exist, here is one.
If x is of the form (2^(n-1)+1)/2^n for some positive integer n, let f(x) be (2^n+1)/2^(n+1).
If x is of the form (2^(n-1) -1)/2^n for some positive integer n, let f(x) be (2^n-1)/2^(n+1).
All other x are sent to themselves.
It is routine to verify that this is a bijection from [0,1] to (0,1). Since there is a bijection between the two sets, they have the same cardinality. (For a homework exercise I would actually do the "routine" bit.)
Cheers,
Ben
"... I couldn't see how anyone could be educated by this self-propagating system in which people pass exams, teach others to pass exams, but nobody knows anything."
--Richard Feynman
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Post #39,753
5/23/02 11:57:17 PM
5/23/02 11:59:16 PM
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Thanks for the quick answers AND...
I'll have to check my notes, but I don't think we could use functions at this point of study. The text, I believe, was Munkres which I think is a pretty standard text. I'll look it up. My memory is that we only had the vaguest of definitions of sets. This was a long time ago, so my memory could be failing, but I think we were artificially prohibited from using functions - as they had not yet been defined.
At any rate, on my first question, do you consider yourself Canadien? And which country do you prefer to identify with? and why?
Thanks again,
Mikem
Edited by mmoffitt
May 23, 2002, 11:59:16 PM EDT
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Post #39,760
5/24/02 12:21:03 AM
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Quick followups
I have heard of Munkres, but I didn't use that text. I don't know what you would have available, but the bijection I gave should be translatable into a straightforward proof no matter what definition was chosen.
As for citizenship, I definitely call myself a Canadian. As for which I identify with, as a practical matter I have more connections to the US, but my heart will always belong to Canada. I wouldn't want to have to choose for real.
Cheers,
Ben
"... I couldn't see how anyone could be educated by this self-propagating system in which people pass exams, teach others to pass exams, but nobody knows anything."
--Richard Feynman
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Post #39,764
5/24/02 12:37:46 AM
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Only because of where you grew up?
I think we all have a predeliction (sp?) to live where we grew up, but I'm wondering if with Canadians (is it really an "a"? - I should stick w/Habs) it isn't more? I've only been to Canada once, and then only briefly to Vancouver. But, if my wife were receptive, I'd move north in a New York minute. Of course, I'd be blinded by my "vacation" perception of Vancouver, but Canada seems like a much more mature, rational place to live than the US. I'd love to hear your take, if you can/care.
Thanks again,
Mikem
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Post #39,872
5/24/02 9:02:27 PM
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Missed saying "I pledge allegiance to the flag..." ;)
Alex
"Men occasionally stumble over the truth, but most of them pick themselves up and hurry off as if nothing had happened." -- Winston Churchill (1874-1965)
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Post #39,904
5/25/02 1:02:59 PM
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dunno, after 20 yrs in AK with only a moving violation
spent 3 weeks in canada and had 3 court appearances on BS. Different parts of Canader have different mores. Ontario is a very much the joneses are watching for round pegs trying to disguise themselves as square and are very quick to get you under the thumb of the authorities for being different. In Quebec things are more laid back especialy the Eastern Townships. In the far north things are good except for the institutional/actual racism towards Natives. Victoria BC is good except for the dam vampires. Dunno about the far east of the country.
my 2 cents,
Bill
TAM ARIS QUAM ARMIPOTENS
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Post #39,907
5/25/02 1:20:17 PM
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Don't know how to answer that
Canada has a rather different attitude towards government, conflict, etc than the US. On the positive, much more acceptance of negotiation, compromise, and generally being nice. The negative side of the same is that bias and bigotry still exist - they are just institutionalized in a way that reduces overt conflict and leaves everyone with plausible deniability.
I am not sure which is better in the end.
But I must admit that I find myself far more often ashamed of my fellow Americans than fellow Canadians. However I find myself outraged at both political systems about equally - though for different reasons.
Cheers,
Ben
"... I couldn't see how anyone could be educated by this self-propagating system in which people pass exams, teach others to pass exams, but nobody knows anything."
--Richard Feynman
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