Re: Sets and common border
- From: "Rupert" <rupertmccallum@xxxxxxxxx>
- Date: 23 Oct 2006 02:29:43 -0700
vred...@xxxxxxxxx wrote:
I was discussing some sets of real numbers in my other post related
some circular algebraic operations, and got into this generic question.
If we take a wooden measuring scale marked with numbers, and break into
parts by cutting it exactly at integer markings (at 1, 2, 3 etc), then
which piece do these numbers belong to?
In other words, if I break a real number line at, say 2, to make it two
sets, then which of the two sets 2 belong to? Here I want to consider 2
as the common border but not an element. Can't we consider numbers as
borders? Then what are borders? Some even thinner points that lie
between each two consecutive numbers?
Venkat
If you partition the real line into two nonempty sets, S and T, such
that for all x in S, if y<x, then y is in S, (that's called a "Dedekind
cut"), then either S={x|x<a} and T={x|x>=a} for some a, or S={x|x<=a}
and T={x|x>a} for some a. This is the completeness property of the real
line that was first observed by Dedekind, he used it to characterize
axiomatically the order type of the real line. Either way, it's
determined which set a is in.
.
- References:
- Sets and common border
- From: vreddyp
- Sets and common border
- Prev by Date: Re: complexity of svds() and eigs() in MATLAB
- Next by Date: New mathematics/physical sciences positions at http://jobs.phds.org, October 23, 2006
- Previous by thread: Sets and common border
- Next by thread: Re: Sets and common border
- Index(es):
Relevant Pages
|
