Talk:Set (mathematics)

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 Definition Informally, any collection of distinct elements. [d] [e]

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, [1], 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)
Isn't there a formal toast attributed to George Boole, "Gentlemen, I give you Pure Mathematics. May it never be of any use to anyone." I believe he died before there was any widespread use of Boolean algebra in digital circuits; I have no data on his rate of rotation in his grave. :-)
Seriously, we are getting into what I consider an important principle of the general knowledge explosion. Believe me, I have no solution, but if anyone has Wikipedia guidelines on watching for it, I'd certainly welcome them.
In this case, I first encountered a horrible example of set theory, called the "new math" when it was a fad in U.S. education...ummm...it was a long time ago. Seventh grade? Eighth grade? Miss Smith's class in junior high school, anyway.
Many years later, I was taking a graduate course in a computer science department, entitled "Discrete Structures". That seems a fairly common advanced undergraduate or graduate course in CS, pulling together an assortment of mathematical disciplines with computer science applications; I've always been amused that the typical undergraduate CS program insists on a year of calculus, for which most programmers find little use (without additional study if they are doing numerical analysis), yet they wait for things like automata theory, and even group theory, which could be useful, in proper context, for first-year students. The physicists had another one-semester course of useful mathematical techniques, none of which made a complete course, for which we don't want our people going to the math department. Useful medical statistics, more often than not, are never really covered in the MD program, but in a supplementary MPH program, or, if one is lucky, in an epidemiology elective.
Anyway, not to digress too much, you make an excellent point that different disciplines talk about similar concepts in different ways. I'm tempted to try to write a short compare-and-contrast section about the issue here, but, while I did some reading of more strictly mathematician's books on set theory (e.g., Hausdorff), I'm sure I don't think as a professional mathematician would (no, I will not wander off into stories about scientists of three disciplines marooned on a desert island...).
Clearly, this has been some free association, but does it give any general ideas? Intuitively, I sense that CZ might be a superb means of detecting cross-discipline usage and finding synergy, but I don't have a good idea how to present it. 10:06, 2 August 2008 (CDT)
An afterthought: should we add Computers Workgroup to the metadata? Even though programmers may not speak of sets very much — architects and designers do — should this article, for example, link to one on data structures and file management, which probably needs to be written? I may do a bit of that, as the article on file transfer program really needs to be redone anyway. Howard C. Berkowitz 10:09, 2 August 2008 (CDT)

The empty set

"The basic property of a set is that it contains elements" Think you have a problem straight away.Gareth Leng 10:12, 2 August 2008 (CDT)

Right, thanks. -- Jitse Niesen 10:43, 2 August 2008 (CDT)

References

  1. Stone, Harold S. (1973), Discrete Mathematical Structures with Applications in Computer Science, Science Research Associates pp. 25-26

Range and codomain

"Range" and "codomain" are not quite synonyms, see WP. Boris Tsirelson 16:43, 10 May 2011 (CDT)

Thank you for pointing out this matter; I hope the change of text is suitable. John R. Brews 19:15, 10 May 2011 (CDT)

family of sets

"A set whose elements are also sets is called a family of sets" -- Yes, sometimes it is called so, to my regret. However, a family (of something) is, by definition, a function from a given index set to these "something". (A family may contain x twice or thrice; a set cannot.) Thus I'd prefer "collection of sets", or "system of sets", while the canonical version is just "set of sets". Boris Tsirelson 00:28, 11 May 2011 (CDT)

Moreover, in the (or at least: a very) usual model of set theory, every object is a set ... --Peter Schmitt 02:49, 11 May 2011 (CDT)
Yes, usual for mathematicians, but not others, I guess. :-) And in practice mathematicians mostly ignore that; for instance sometimes we say "set function", but surely not about (say) a polynomial... Boris Tsirelson 02:57, 11 May 2011 (CDT)
OK, I have tried to take these comments into account, but am not sure I have made it. Please take a look and revise if you like. John R. Brews 04:33, 11 May 2011 (CDT)

Set of sets

"(It may be observed that if a set is defined by a property, not by enumeration, there may be some complexity in discussing its subsets.)" -- I am puzzled: what is meant? --Boris Tsirelson 03:12, 13 May 2011 (CDT)

Hi Boris: What I meant to say here was that while a set defined by simply listing its members lent itself readily to identifying its subsets by identifying their members, a set defined by a rule and not by listing its elements would require its subsets to be identified by a rule as well, and then you are faced with the issue of proving that the subset defined by its rule was indeed a subset of another set defined by another rule. Of course, that may be easy to do, but it appeared to me that it also might be a subtle matter, depending upon the rules involved. John R. Brews 07:52, 13 May 2011 (CDT)
I see. If by "a set defined by simply listing its members" you mean a finite set, then of course, it is much simpler notion than a set in general. And not only w.r.t. subsets, but also in all other aspects. But if "1,2,3,..." is also treated as a (infinite) list, then its subsets can be quite problematic. Anyway, if such a remark is needed in the article then it should be made reasonably clear to the reader. --Boris Tsirelson 08:41, 13 May 2011 (CDT)
I've rewritten this sentence. John R. Brews 09:28, 13 May 2011 (CDT)