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[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 XSL-List info and archive: http://www.mulberrytech.com/xsl/xsl-list
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