Talk:Unordered pair

Requested move

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: page moved. Vegaswikian (talk) 03:48, 2 April 2010 (UTC)

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Binary set → Unordered pair — Change to much more common name Hans Adler 16:05, 26 March 2010 (UTC)

I could think of four alternative names for sets with 2 elements and checked their frequency in Google Books by searching only in books that have "set theory" in the title to get rid of spurious results. Here is the result:

• 79x unordered pair
• 38x two-element set
• 23x pair set
• 3x binary set, but these hits referred to three different unrelated things (binary set operation, binary set theory, binary set-relation).

The move seems to be a no-brainer, but the target has trivial history. Hans Adler 16:05, 26 March 2010 (UTC)

• Agree seems to be the most descriptive name. --Salix (talk): 19:48, 26 March 2010 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Unordered pairs and binary sets are different and distinct concepts

This page has become very confused. It's first sentence "an unordered pair is a set that has exactly two elements" is simply wrong. A set with cardinality 2 must have two distinct elements; the elements of an unordered pair need not be distinct. So (3,3) can denote an unordered pair, but it cannot denote a set with two elements - as a set, it only has one element.

Now you could, perhaps, define a correspondence between sets with one or two elements and unordered pairs, in which the set {a,b} (a and b distinct) corresponds to the unordered pair (a,b) and the set {a} corresponds to the unordered pair (a,a). But I think it would be difficult to extend that definition to unordered triples etc. Seeme to me it would be more natural to define an unordered pair as an equivalence class of ordered pairs. The unordered pair (a,b) (a and b distinct) is then the set {(a,b), (b,a)} containing two ordered pairs, whereas the unordered pair (a,a) is the set {(a,a)} containing a single ordered pair. There are then various ways of defining ordered pairs in terms of sets (see the ordered pair article).

I don't have a source for this to hand, so I will leave it to others to fix the article with an appropriate source. There may be other "right" ways of defining an unordered pair, but the article as it stands is clearly not correct.

Incidentally, in this version, before the March 26 re-write and the subsequent renaming, the article gave an almost correct definition of a binary set, which was its title at the time. The rewrite and renaming have conflated the concepts of binary set and unordered pair, and have turned a good article into a very poor one. Gandalf61 (talk) 09:43, 3 April 2010 (UTC)

I am travelling and don't have access to my books right now, but I am very confident that what this article describes is a major interpretation of the term "unordered pair". I would also guess that it is the most common one and possibly the most common one by far. But perhaps my POV is skewed by the fact that I only know precise definitions of the term from set-theoretic contexts? Do you have some other context in mind that works with unordered pairs in your sense? Hans Adler 20:50, 3 April 2010 (UTC)

Well, an ordered pair is an ordered 2-tuple, so when I see "unordered pair" I assume it means an unordered 2-tuple i.e. an equivalence class of ordered 2-tuples. Thus the unordered pair (1,2) is the equivalence class {(1,2),(2,1)} containing two ordered 2-tuples, whereas the ordered pair (3,3) is the equivalence class {(3,3)} containing a single ordered 2-tuple. As it stands, the article appears to deny that (3,3) can be an unordered pair at all ("if a = b, then {a, b} = {a}, and the set is not an unordered pair but a singleton"), which seems to me to be bizarre. Gandalf61 (talk) 13:36, 4 April 2010 (UTC)

I don't think it's bizarre at all. If you look at mathematical literature from the early 20th century, when people were not familiar with set theory yet (or at least didn't use it as a language), you will see that they typically talked about just "a pair a, b". Sometimes the implicit order between a and b mattered and sometimes it didn't matter. Very often it was implicitly assumed that a and b were distinct. Sometimes it was said explicitly that they were allowed to be equal.

When mathematics became more formal, the term pair was split into the ordered pair (a, b) and the unordered pair {a, b}. For the former it was convenient to permit a = b, for the latter it was (apparently) more convenient not to permit this. After a while, the word ordered was dropped and ordered pairs became just pairs. This is all original research, based on my (somewhat limited) familiarity with older mathematical texts. When I get home I will try to locate books that make this explicit. Whether I am right or not, if we can find an explanation of the historical development of this terminology it would be very valuable for this article. Hans Adler 14:29, 4 April 2010 (UTC)

After a quick search through Google Books and Google Scholar I am beginning to agree with you. Just give me a day or two to check whether my POV also appears in the literature; after that I will fix the article. I have just found some interesting historical literature about implementation of ordered pairs in set theory, so in the end our articles in this area will be improved under all circumstances. Hans Adler 15:00, 4 April 2010 (UTC)

Now I have more information:

• A large majority of set theory texts defines unordered pairs in such a way that any set of the form {a, b}, i.e. any set with one or two elements, is called an unordered pair.
• I have not found a single source that defines unordered pairs as equivalence classes (permutation orbits) of ordered pairs.
• I have found three sources that define an unordered pair as any set of the form {a, b}, but also an unordered triple as any set of the form {a, b, c}. This is not equivalent to defining it as an equivalence class of ordered triples, since (a, a, b) and (a, b, b) are not the same up to permutation (if a ≠ b), but induce the same set:
  • Hrbacek, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 978-0-8247-7915-3.
  • Rubin, Jean E. (1967), Set theory for the mathematician, Holden-Day.
  • Takeuti, Gaisi; Zaring, Wilson M. (1971), Introduction to axiomatic set theory, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag.
• I have found four sources that define an unordered pair as a set of the form {a, b} with a ≠ b, or as a set with exactly two elements:
  • Düntsch, Ivo; Gediga, Günther (2000), Sets, Relations, Functions, Primers Series, Methodos, ISBN 978-1-903280-00-3.
  • Fraenkel, Adolf (1928), Einleitung in die Mengenlehre, Berlin, New York: Springer-Verlag.
  • Roitman, Judith (1990), Introduction to modern set theory, New York: John Wiley & Sons, ISBN 978-0-471-63519-2.
  • Schimmerling, Ernest (2008), Undergraduate set theory.

Note that the sources defining an unordered pair as a two-element set are generally of relatively low quality (two are unpublished/self-published, three are for undergraduates), except for the very early Fraenkel (1928), which discusses the question in some detail (my translation):

Axiom II. If a and b are distinct sets, then there is a set which contains the elements a and b but no element distinct from them. This set is to be described by {a, b} and is called the pair of a and b. (Pairing axiom.)

[...]

I we left out the word "distinct" in the assumption of the axiom, the existence of the pair (which, however, would no longer deserve this designation) would be required also in the case a = b; the axiom would therefore contain a stronger requirement. But the above weaker and therefore simpler version is sufficient for our purposes.

Fraenkel uses the old terminology "ordered pair"/"pair", rather than the new terminology "[ordered] pair"/"unordered pair", so where he writes "Paar" (pair) he actually means unordered pair.

I think what happened is probably this:

• "Pair" branched into "ordered pair" and "unordered pair".
• An "unordered pair" was (and is) a set. Originally it had precisely two elements for most (or at least many) authors.
• In written form "unordered pairs" came up (under this name) almost exclusively in the pairing axiom that says that for any two sets a and b, {a, b} is a set.
• In the context of this axiom the restriction a ≠ b does not make much sense: It makes the axiom more complicated (although Fraenkel claimed the opposite), and the existence of singleton sets can either be deduced using other axioms or needs to be postulated additionally. Therefore the axiom was made stronger, but its name ("pairing axiom", "existence of unordered pairs") stayed the same.
• In most texts the axiom is the only place where "unordered pairs" are defined, so they are defined without the restriction.

I think this justifies defining unordered pairs as all sets of the form {a, b}, but with a warning that some authors may require a ≠ b. Hans Adler 21:35, 6 April 2010 (UTC)