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Grex > Tutoring > #14: Algebra, Geometry, Calculus, Trig, all that good stuff |  |
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| 20 new of 132 responses total. |
i
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response 113 of 132:
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Sep 4 22:52 UTC 1997 |
(Compared to the actual growth rates, 25% per generation [less early
mortality, infertility, etc.] would have been wonderfully low, and
India would be in much better shape today.)
Steve's "H O W E V E R..." in #111 is the correct and "best" solution.
There are simpler "add things up" solutions, but this is one of those
problems where the student's approach tells the teacher more about his/her
understanding than anything else.
(The instructor threw this problem at a class of PhD-track math (not stats)
grad students I was in. The initial class consensus was for a wrong answer
[and they almost agreed on which one].)
On to a more familiar mathematical topic - Sets. Apples and oranges are
elements of the set of fruits, Mr. Figston's 3rd grade is a subset of the
set of students at Washington Elementary School, the intersection of the
set of wet thing with the set of dry things is the empty set, and all that
fun.
One very popular set is the Universal Set, which contains EVERYTHING - real,
abstract, imagined or undiscovered, simple or complex, it's all there. But
the idea that such a set can exist suffers from a fatal logical flaw. It
is not that the Universal Set can't contain itself as a subset - but that's
a good hint on where to start looking.
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remmers
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response 114 of 132:
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Sep 5 00:34 UTC 1997 |
Did you mean to say "contain itself as a subset" or "contain
itself as an element"?
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srw
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response 115 of 132:
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Sep 5 02:38 UTC 1997 |
resp:112 Hey, that's cool, Mark. I learned how to solve that silly
series using differentiation. Your way is much better.
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tpryan
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response 116 of 132:
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Sep 5 22:37 UTC 1997 |
Would the Universal Set, then try to contain 'null' and 'infinity'
at the same time?
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i
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response 117 of 132:
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Sep 5 22:59 UTC 1997 |
The Universal Set (U for short), by "definition", contains null, infinity,
and anything else you can think up. Yourself included.
Re #114: U must do both. I believe that a contradiction can be derived
either way, but the "nontraditional" one requires minimal knowledge of
power sets.
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remmers
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response 118 of 132:
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Sep 6 13:15 UTC 1997 |
Well, every set contains itself as a subset, so I figured you
must have meant "contain itself as an element".
The most familiar contradiction based on the notion of sets
containing themselves as elements is the Russell Paradox, which
goes as follows: Let S be the set of all sets that are not
elements of themselves. Then if S is an element of itself, then
by definition of S, S is not an element of itself. Conversely,
if S is not an element of itself, then again by definition of
S, S is an element of itself.
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rcurl
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response 119 of 132:
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Sep 6 16:09 UTC 1997 |
Mike shaves all men that do not shave themselves. Does Mike shave himself?
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mcnally
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response 120 of 132:
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Sep 6 16:40 UTC 1997 |
re #119: Your statement of Russell's "Barber" paradox is insufficient
to indicate the paradox since it says nothing about whether or not the
barber shaves some of those who shave themselves (which would only include
the barber..) or whether or not the restriction applies to the barber
(e.g. is the barber a woman?) You need a restriction more like "the barber
shaves all those and *only* those who don't shave themselves.."
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rcurl
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response 121 of 132:
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Sep 6 17:42 UTC 1997 |
Ok, fine. But if you are going to be particular, remember that the
defining relative pronoun is *that*, so it has to be stated as "the barber
shaves all those and only those *that* don't shave themselves."
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i
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response 122 of 132:
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Sep 7 14:06 UTC 1997 |
Re: #118 - my meaning was that a contradiction can be obtained by looking
either at sets which are elements of U or at sets which are subsets of U
(the power sets are used in the latter). I did't want to give too big a hint.
This Statement Is False.
(Is the above statement a paradox? Is it not a paradox only for "poetic"
reasons - the contradiction is too poorly hidden, insufficiently interesting,
etc.? What is a paradox?)
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tpryan
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response 123 of 132:
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Sep 12 23:00 UTC 1997 |
Hey, I know Barry & Sally Childs-Helton, they both have Phds,
they are a Paradox.
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dang
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response 124 of 132:
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Sep 29 17:16 UTC 1997 |
re 122: two places to moor ships.
Speaking of Russel's paradox, Greg (my roommate, flem) was just reading his
autobiography and mentioned it to me last night. Interesting coincidance.
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lilmo
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response 125 of 132:
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Dec 18 01:23 UTC 1997 |
Isn't a paradox something that appears to be true, even though it is not?
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srw
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response 126 of 132:
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Dec 19 03:40 UTC 1997 |
Er, not exactly. I would describe a paradox as an assertion that seems to be
contradictory. Almost the opposite of your definition.
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rcurl
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response 127 of 132:
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Dec 19 17:40 UTC 1997 |
I agree with Steve. A paradox appears to be internally contradictory.
Paradoxes may have resolutions, or they may not. Something that "appears to
be true, even though it is not" is just an error. I think there is
an expression for it, such as "plausible but mistaken".
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lilmo
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response 128 of 132:
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Jan 3 21:56 UTC 1998 |
Chalk one up to fuzzy-headedness. *sigh*
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anilkk
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response 129 of 132:
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Jan 13 17:41 UTC 1999 |
paradox is self-contradictory.
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lilmo
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response 130 of 132:
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Apr 17 00:02 UTC 1999 |
Re #129: A paradox is something that *appears* so be self-contradictory.
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rcurl
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response 131 of 132:
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Apr 17 04:01 UTC 1999 |
That's what I said in #127....... ;)
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lilmo
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response 132 of 132:
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Apr 20 01:08 UTC 1999 |
Apparently, he missed it the first time. :-)
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