[xsl] sets and ordering (semi-off-topic)

Subject: [xsl] sets and ordering (semi-off-topic)
From: "Dave Gomboc" <dave@xxxxxxxxxxxxxx>
Date: Thu, 25 Jan 2001 14:56:05 -0700
I sent this (with small changes) privately earlier, but I just received
a digest with about eight posts on this topic, so there appears to be
sufficient interest on the list for me to post it publicly.  While I'm
at it, I'd also like to take this moment to apologise for the email that
I unintentionally sent to the list yesterday.  Sorry! :-)

An ordering over a set is defined by a relation, which actually is a
different set.  Sets themselves are _always_ unordered, even the ones
that express relations.

For example, the relation < (less than) over R (the real numbers) is the
set of all ordered pairs of real numbers such that the first is less
than the second.  [Of course, you need an ordered pair to tell the first
from the second: what we would commonly write as (1,2) would be
expressed by {1, {1, 2}} in set notation.]

One can speak of the set (R, <), but this actually refers to the set
containing elements such as {1, {1, 2}} and {1, {1, 3}}, not the set of
real numbers.

A good reference for this stuff (e.g. where I looked it up :-) is
Elements of the Theory of Computation, by Lewis and Papadimitriou
(Prentice-Hall, 1981), chapter 1, pages 5-8.

I'll leave it to others to discuss XSL's node sets and lists.  I just
wanted to clarify the meaning of the noun "set" as used in mathematics
and computing science.

Dave Gomboc

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