Timeline for Confusing notation for sets of unordered vs ordered pairs
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 7, 2021 at 17:29 | comment | added | Federico Poloni | I see a lot of content here that should have been in answers and not in comments. | |
| Jan 7, 2021 at 17:04 | comment | added | Aditya Guha Roy | You can find some more definitions here en.wikipedia.org/wiki/Ordered_pair . | |
| Jan 7, 2021 at 16:49 | comment | added | Matthieu Latapy | Also, I agree that unordered pairs make sense mostly (if not only) when $X\cap Y \neq \emptyset$. | |
| Jan 7, 2021 at 16:48 | comment | added | Matthieu Latapy | Thanks for your interesting comments! It seems that there is no consensus, rather a nice bestiary of relevant notations, each with its own strengths and weaknesses. | |
| Jan 7, 2021 at 16:44 | comment | added | Aditya Guha Roy | Regarding notations there is some room for confusion unless one is not very clear about them. In context graph theory, I personally feel that contrasting the relationship between ordered and unordered pairs with that between directed and undirected graphs is helpful (assigning an order to a set can be contrasted with assigning a direction to traverse the elements of the set); and so if you are consistent about any specific notations to distinguish directed graphs from undirected ones, you can adopt the same here. I personally write $\vec{G}$ to denote directed graphs in some cases. | |
| Jan 7, 2021 at 16:37 | comment | added | Aditya Guha Roy | Ordered pairs (or in general ordered sets) have a set theoretic definition as an unordered set. While the definition is most likely to depend on the purpose, people mostly use ordered pairs to distinguish $(x,y)$ from $(y,x).$ There are multiple ways to define $(x,y)$ so that it has this property, for instance one can define $(x,y) = \{ x, \{x,y \} \}$ (this was actually coined by Kuratowski). The existence of ordered sets can then be verified as a consequence of ZF axioms. So, you can view an ordered set as an unordered set too, if that helps! | |
| Jan 7, 2021 at 14:23 | comment | added | Francesco Polizzi | If $X \neq Y$, I am not sure that the notion of "unordered pair" really makes sense, because one does not have the $\mathbb{Z}_2$-action anymore. | |
| Jan 7, 2021 at 14:18 | comment | added | Francesco Polizzi | The usual notation is $\mathrm{Sym}^2(X)$, the second symmetric product of $X$. It can be seen as the quotient of $X \times X$ by the $\mathbb{Z}_2$-action exchanging the two factors. | |
| Jan 7, 2021 at 14:16 | comment | added | Dan Petersen | @Qiaochu writes that he would not consider the set of unordered pairs from two sets $X,Y$ meaningful unless $X = Y$. I would rather say that the notion makes sense precisely when $X$ and $Y$ are both subsets of an ambient superset $T$, so that the intersection $X \cap Y$ is meaningful. But if $X$ and $Y$ are arbitrary sets then I agree that the notion makes little sense. A verbose way of expressing this operation is $\mathrm{Im}(X \times Y \to \binom{T}{2})$. A modest proposal for a notation is $X \, \Box \, Y$ (I am quite convinced no standard notation exists). | |
| Jan 7, 2021 at 13:07 | comment | added | Noah Schweber | @QiaochuYuan The notation "$[X]^2$" for the set of two-element subsets of $X$ is standard at least in logic. | |
| Jan 7, 2021 at 12:29 | comment | added | Matthieu Latapy | I like $X\choose 2$, but am disturbed by the fact that it does not generalize to two distinct sets. Also, I am not that happy with the combined use of both notations, like, e.g., $T\times {X\choose 2}$. I have similar concerns with $X^{(2)}$. | |
| Jan 7, 2021 at 12:29 | comment | added | Matthieu Latapy | Thank you for these interesting comments. Indeed, there cannot be any repetition in an unordered pair, as it is a set, but I saw that some consider $\{x\}$ as a valid unordered pair so I wanted to be precise. | |
| Jan 7, 2021 at 11:30 | comment | added | Pietro Majer | In fact the usual convention is that $\{x\}=\{x\}$, so that one does not have to bother about repetitions. As to the set of unordered pairs, $X\times X$ is of course a totally wrong notation; $X\otimes X$ is at least different from $X\times X$, but confusing in that it has nothing to do with any other use of $\oplus$. ${X \choose 2}$ follows the standard convention about cardinality, it is self-explanatory and natural. Also used $X^{(n)}$ for unordered $n$-ples | |
| Jan 7, 2021 at 10:39 | history | edited | Matthieu Latapy | CC BY-SA 4.0 |
added 1 character in body
|
| Jan 7, 2021 at 10:34 | comment | added | Qiaochu Yuan | I would not consider the set of unordered pairs from two sets $X, Y$ meaningful unless $X = Y$, in which case I would write it as ${X \choose 2}$ (assuming we want distinct pairs). | |
| Jan 7, 2021 at 9:31 | history | asked | Matthieu Latapy | CC BY-SA 4.0 |