Size of Infinite Sets

Date: 11/20/96 at 21:01:17
From: Gary Zak
Subject: Conics and Infinity

Which is larger: the number of parabolas, or the number of hyperbolas?

For any given double-napped cone, consider a plane surface 
intersecting the cone, parallel to a generatrix, but not through the 
vertex of the cone. The points of intersection between the plane 
surface and the cone form a shape called a parabola. This parabola 
will have its own vertex, a point on the cone. Through this point 
there can be one and only one parabola. However, plane surfaces that 
share this vertex, but are not parallel to that generatrix can 
generate many hyperbolas - in fact, an infinite number of them. This 
line of reasoning suggests that there are more hyperbolas than 
parabolas.

However, since there are an infinite number of parabolas on any given 
double-napped cone (by varying the distance of the plane from the 
vertex of the cone), is it not possible to map on a one-to-one basis 
each hyperbola to a parabola?  This line of reasoning suggests that 
there are an equal number of parabolas and hyperbolas, i.e. an 
infinite number of each. 

The next consideration is whether the infinite number of possible 
parabolas is of a different nature than the infinite number of 
hyperbolas. Since I do not have enough of a high level math background 
to "horse around" with this myself, I am seeking more opinions and/or 
facts. I suspect I will need to approach this problem at both a Gallup 
and a Cantor!

Thanks in advance for any replies,

Gary Zak

Date: 11/21/96 at 14:31:29
From: Doctor Tom
Subject: Re: Conics and Infinity

Hi Gary,

Actually, although "a Gallup and a Cantor" may shed some light on the 
problem, I think Lebesgue may provide the answer.  Lebesgue was one of 
the founders of the field currently known as "measure theory".  A math 
major will usually run into it in the junior or senior year, and then 
in a more detailed course in real analysis in graduate school.

There are some tricky ideas when you're trying to compare the sizes of 
infinite sets.  Cantor just helps you deal with the cardinality of the 
sets -- whether they can be put in one-to-one correspondence.

In fact, from Cantor's point of view, there are exactly the same 
number of parabolas as hyperbolas (the same cardinality), but in any 
reasonable assignment of "measure" to the sets, the measure of the set 
of parabolas will be zero compared to the set of hyperbolas.

Here's a "parable" that may help you see what's going on.  Consider
a unit square on the plane, and the diagonal line of that square.  
Both (according to Cantor) have the same cardinality: C, where "C" is
the cardinality of the real numbers, and C^2 = C (in transfinite
arithmetic).

But the area of the square is 1 and the area of the diagonal line is
zero.  So if you threw darts with an infinitely tiny point at the 
square, the chance that they would land somewhere in the square is 
1.0, and the chance that they would hit the line exactly is 0.0.

Consider all conic sections.  They are described by a 5 parameter
family:

   A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0

There are 6 variables: A, B, C, D, E, and F, but one isn't really 
there because you can multiply by a constant.  Although this isn't
exactly right, I'll just divide through by F, so let F=1 and A through 
E are the 5 parameters.  This isn't strictly kosher, since F might 
have been zero, but let's ignore that for now.

So consider any region in any 5-dimensional "block" with the variables 
A, B, C, D, and E.  For some selections of these 5 numbers, the 
equation represents a parabola, for others a hyperbola, and for still 
others, an ellipse.  The regions corresponding to hyperbolas and 
ellipses represent full 5-dimensional regions, but the parabolic 
regions are represented by a 4-dimensional surface in that block.  
The 5-dimensional volume of a 4-dimensional surface is zero (just as 
the 2-dimensional "volume" of a line is zero in my parable).

The chance of a 5-dimensional "dart" hitting a point corresponding to 
a parabola is zero.

By the way, certain degenerate conics, such as crossed lines, points
(x^2 + y^2 = 0), et cetera, have even lower dimensionality than
parabolas, and in that sense are "infinitely less" probable than
parabolas.

If you want to know more about Cantor, look at this web site:

  http://www.shu.edu/~wachsmut/reals/history/   

I hope this helps.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/