Talk:Set (mathematics)

Paradoxes, ordered sets

In the beginning, a set is described in an axiomatic way, without a rigorous definition. Have you thought about text that avoids Russell's Paradox? http://plato.stanford.edu/entries/russell-paradox/

I came to the article because I wanted to link to "ordered set". Is that one of the special sets here, should there be a section for it, or should there be a new article? For that matter, should this refer to or define tuples?

Howard C. Berkowitz 11:06, 28 July 2008 (CDT)

I wouldn't say it's described in an axiomatic way, because no axioms are mentioned. The text only hints that sets in mathematics are described axiomatically. The axioms that are used nowadays (usually ZFC) avoid Russell's paradox, but these axioms are rather complicated to explain; see http://eom.springer.de/Z/z130100.htm .

I'm not quite sure what you mean by "ordered set", but I guess it would be usually be called "sequence" or "tuple" by mathematicians. The concept of an ordering is not mentioned in the article, but it should be. At least there should be a link to sequence or tuple. -- Jitse Niesen 11:04, 29 July 2008 (CDT)

Tuples and ordered sets

:-) I'm glad you phrased that as "mathemeticians", as "computer scientists" sometimes come up with shortcut terms that apply to real-world examples, as much as software is, or is not, real world. I'll start with CS analogies, and then go to more formal definitions.

The degenerate case of a tuple is a single scalar or boolean that can be an element of a set S. More usefully, tuples T are structured groups of elements, such as <a,b> or, in pseudo-C (assuming Boolean type has been defined) that I don't have to format too much here,

struct T{

       boolean major;

       boolean minor;

}

In more formal notation, T is an ordered pair or a 2-tuple. As a programmer, I might think of a file FS as having a sorted sequence of records that are ordered pairs.

Having pulled out my CS graduate school text, , orderings are defined as relations with certain properties. If I set up an ordering relations for an ascending sort, FS ={<0,0>,<0,1>,<1,0>,<1,1>}

Stone doesn't have "tuple" in that section. Howard C. Berkowitz 14:59, 30 July 2008 (CDT)

I added a bit to the article explaining the differences between sets and tuples. Please have a look and see whether it makes sense to you, and improve it if you can.

Mathematicians do talk about "ordered sets", but it means something slightly different. Like a set, an ordered set can contain an element only once. The only ordered sets containing two booleans are (0, 1) and (1, 0). In contrast, there are four pairs (or 2-tuples) containing booleans, namely <0,0>, <0,1>,<1,0>, and <1,1> (as you said). That's the difference between ordered sets and tuples in mathematics.

Of course, when I say "mathematicians call …", I don't want to imply that mathematicians are better than computer scientists. It's just that different fields often use different terminologies. -- Jitse Niesen 09:32, 2 August 2008 (CDT)