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A function that turns a real number into another real number can be represented like $f : \mathbb{R}\to \mathbb{R}$

What is the analogous way to represent a function that turns an unordered pair of elements of positive integers each in $\{1,...,n\}$ into a real number? I guess it would almost be something like $$f : \{1,...,n\} \times \{1,...,n\} \to \mathbb{R}$$ but is there a better notation that is more concise and that has the unorderedness?

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I have seen $[n]$ for $\{1,2,3,\ldots n\}$ but it is always defined, not considered standard like $\mathbb{R}$. You could extend your function to be on ordered pairs by symmetry, but maybe that obscures a point you want to make. – Ross Millikan Oct 21 '11 at 16:04
you may also consider set of unordered pairs as a triangle $X = \{(i,j): 1 \leq i\leq j \leq n\}$ but it maybe also obscuring. – Ilya Oct 21 '11 at 16:23
If $X$ is any set you may denote the set of unordered pairs of elements of $X$ by ${X\choose2}$. So your function can be described as $f:\ {[n]\choose 2}\to{\mathbb R}$. – Christian Blatter Oct 21 '11 at 19:46
@Christian: I generally see ${X \choose 2}$ used to denote the set of subsets of $X$ of size $2$, which is very close but not quite the same thing. – Qiaochu Yuan Oct 21 '11 at 20:23
@Qiaochu Yuan: I don't know whether you can call a singleton $\{a\}$ an "unordered pair". The OP will have to decide what he actually meant. – Christian Blatter Oct 22 '11 at 20:01

I would say that it might be best to preface your notation with a sentence explaining it, which will allow the notation itself to be more compact, and generally increase the understanding of the reader. For example, we could write:

Let $X=\{x\in\mathbb{N}\mid x\leq N\}$, and let $\sim$ be an equivalence relation on $X^2$ defined by $(a,b)\sim(c,d)$ iff either $a=c$ and $b=d$, or $a=d$ and $b=c$. Let $Y=X^2/\sim$, and let $f:Y\to\mathbb{R}$.

So, $Y$ can be thought of as the set of unordered pairs of positive integers up to $N$, and you can then proceed to use this notation every time you want to talk about such a function.

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I gave a +1 for your first sentence. – J. M. Oct 22 '11 at 1:49

The set $\{1, \ldots ,N \}$ is often written as $[N]$, so this could be $f: \operatorname{Sym}^2([N]) \to \mathbb{R} $. Here $\operatorname{Sym}$ means the symmetric product, that is, $\operatorname{Sym}^2(S)$ can be thought of as the set of unordered pairs of elements of $S$.

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interesting, I didn't know about either the [N] notation or the $Sym^2$ notation. – opt Oct 21 '11 at 16:17
The $Sym$ notation is usually used with vector spaces, sometimes in conjuction with wedge notation (wedge is the anti-symmetric product). Using it with sets is a bit of abuse of notation. – Craig Oct 21 '11 at 17:51

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