Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$.
One may also consider unordered pairs $xy$ with $x\in X$ and $y \in Y$, with $x\ne y$. Then, $\{x,y\}=\{y,x\}$, and $\{x,x\}$ does not exist.
The set of all ordered pairs is denoted by $X\times Y$, and the set of unordered pairs is often denoted by $X\times Y$ too, which is confusing.
For instance, in graph theory ordered vs unordered pairs make the distinction between directed and undirected graphs. Therefore, most graph papers dealing with undirected graphs start by saying that they use the ordered pair notations, but consider unordered pairs. In most cases, this is sufficient.
In some situations, though, one deals with both kinds of pairs jointly and then needs to make a clear distinction.
It was for instance the case in our paper Stream Graphs and Link Streams for the Modeling of Interactions over Time, where we used the $X\otimes Y$ notation for unordered pairs.
My questions are:
- is there a standard notation for sets of unordered pairs? Why not ?
- And is the $\otimes$ notation appropriate, although it is already used in various contexts?
A typical example where the distinction is important, although trivial: $|X\times Y| = |X|\cdot |Y|$, and so $|X\times X| = n^2$ if $|X| = n$. This is different from $|X\otimes Y| = |(X\setminus Y) \times Y| + |(Y\setminus X) \times X| - |(X\setminus Y)\times(Y\setminus X)| + \frac{|X\cap Y|^2 - |X\cap Y|}{2}$,$$|X\otimes Y| = |(X\setminus Y) \times Y| + |(Y\setminus X) \times X| - |(X\setminus Y)\times(Y\setminus X)| + \frac{|X\cap Y|^2 - |X\cap Y|}{2}$$ leading to $|X\otimes X| = \frac{n \cdot (n-1)}{2}$ if $|X| = n$.