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[Watno]

There's also Discrete Mathematics.

Why is Set Theory associated with Algebra by the way? My university does this as well, but I don't really see it (though I haven't done an set theory lectures yet).

[SirPeebles]

I don't usually see algebra lumped with set theory.

[sudgy]

Hehe, I just read about power series and such last night, and now understand SirPeebles first thing about it.  I'm trying to find a good smiley for this but I can't think of it...

[soulnet]

There's also Discrete Mathematics.

Why is Set Theory associated with Algebra by the way? My university does this as well, but I don't really see it (though I haven't done an set theory lectures yet).

You can see any logic as an algebra.

(I am assuming by "Set Theory" you mean something like "ZFC theory" or its vicinity)

[pacovf]

You can see anything as a category, if you think about it long enough.

EDIT: your head may explode before that, though.

[soulnet]

You can see anything as a category, if you think about it long enough.

ugh, categories. That is just the extreme result of the fallacy "more general is better". Using too general tools usually hides the meaning of what you prove.

[SirPeebles]

You can see anything as a category, if you think about it long enough.

ugh, categories. That is just the extreme result of the fallacy "more general is better". Using too general tools usually hides the meaning of what you prove.

I've got to disagree here.  Category theory opens up some incredibly beautiful unifications and dualities. 

[soulnet]

You can see anything as a category, if you think about it long enough.

ugh, categories. That is just the extreme result of the fallacy "more general is better". Using too general tools usually hides the meaning of what you prove.

I've got to disagree here.  Category theory opens up some incredibly beautiful unifications and dualities.

That's not really disagreeing. My point is not "category theory is useless" but "trying to get category theory to be an all-encompassing theory" is a counterproductive goal, regardless on whether it is possible or not, and a significant amount of category theorist seem to advocate that goal (this is hugely biased by the ones that I know or have heard off, and not being in the field myself, could easily be a narrow view).

[navical]

I think that proof works. Nice. Also, yes, I probably should have specified that no three points are collinear.

Here is mine:


Consider 4 points which are the endpoints of two intersecting line segments. Those points form a convex quadrilateral, and the segments are the diagonals of that quadrilateral. We can re-pair those 4 points so that those two segments are two opposite sides of the quadrilateral and do not intersect. From the triangle inequalities, we know that the sum of the lengths of two opposite sides of a convex quadrilateral is strictly less than the sum of the lengths of the diagonals.

Now, consider the pairing of points which leads to the least total length of the line segments. We know that such a pairing exists since there is a finite number of points. Assume that two of the segments in this pairing intersect. Then, we can re-pair them as described above, and the total length of the line segments will decrease. But that is a contradiction, since we started with the pairing with the least total length.

Therefore the pairing with the least total length of line segments will not have any intersecting segments.

Oh, very nice.

[qmech]

Here's another: is it always possible to cover 10 marks on a white table cloth with 10 plates that cannot overlap?

Reposting this.  The solution is a very neat trick which is tremendously useful when doing real maths.

[SirPeebles]

Here's another: is it always possible to cover 10 marks on a white table cloth with 10 plates that cannot overlap?

Reposting this.  The solution is a very neat trick which is tremendously useful when doing real maths.

Surely it depends on the sizes and shapes of the plates and of the cloth.

[qmech]

Here's another: is it always possible to cover 10 marks on a white table cloth with 10 plates that cannot overlap?

Reposting this.  The solution is a very neat trick which is tremendously useful when doing real maths.

Surely it depends on the sizes and shapes of the plates and of the cloth.

Marks are points and plates are disks of equal size.  Assume the tablecloth is R^2.  The question is whether it's possible to cover all of the points with any number of disks (although you obviously don't need more than 10).

Note that it's important what number 10 is: if I can put points sufficiently densely everywhere then it's impossible because the disks can't overlap, and it's obviously possible for, say, 2 points.

[pacovf]

I assume you can choose the size of the plates?

But then if you can choose the size of the plates, the answer is obviously yes, you just have to get small enough plates.

And if you can't choose the size of the plates, you are not even sure that you can fit all 10 plates on the cloth, so the answer is no, you can't always fit them in such a way as to cover all 10 points.

I am with SirPeebles in this one, I think we are missing a premise.

[qmech]

I assume you can choose the size of the plates?

No.  Or yes, but the points are placed afterwards.

[pacovf]

I assume you can choose the size of the plates?

No.  Or yes, but the points are placed afterwards.

Huh? If the points are placed after placing the plates, you can always place them outside the plates, since the plates are disks...?

[qmech]

The disks are placed after the points.

[Watno]

I think the suggested sequence was
1)Choose a size for the plates
2)Place points
3)Place plates

Also he said that the cloth is R^2

[liopoil]

must the center of each plate be over the tablecloth? Or can you hang them over the table, with just a little bit on the table.

[Watno]

Also he said that the cloth is R^2

[qmech]

must the center of each plate be over the tablecloth? Or can you hang them over the table, with just a little bit on the table.

That would be allowed: you can assume the tablecloth is infinite.

[liopoil]

must the center of each plate be over the tablecloth? Or can you hang them over the table, with just a little bit on the table.

That would be allowed: you can assume the tablecloth is infinite.

...then how is the tablecloth R^2?

[liopoil]

Am I missing what R^2 means?

[Watno]

R^2 is supposed to be the plane of real numbers

[qmech]

Sorry, R^2 is what I call an infinite (two-dimensional) plane, because you can specify any point in it with two real numbers.  Just think of it as an infinitely large tablecloth.

[liopoil]

oh, duh, yeah, I've seen a capital R used for the set of real numbers before, although normally it's supposed to be all fancy and stuff, right? But that's inconvenient to type of course. Anyway, yeah, I don't know how to go about proving that. I feel like there should be a fairly simple iterative process to find a configuration. Although maybe it's possible to show that there is a configuration without finding it. Well, I'll read someone's solution whenever they figure it out .