6
$\begingroup$

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?

  • 0
    @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.2011-10-22

2 Answers 2

8

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.

  • 0
    I gave a +1 for your first sentence.2011-10-22
7

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 $S$.

  • 2
    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.2011-10-21