Binary Function on an unordered pair?

Question

We define binary functions such as addition over the reals can be defined like this in the various notations, $+(a,b)=a+b=b+a=+(b,a)$ in this case there is a separate mapping as for the tuples $(a,b)$ and $(b,a)$ for $a,b$ in $R$, can we define the addition function so that it acts on an unordered pair and therefore has one mapping for the sum of $a$ and $b$, and therefore for the unordered pair $[a,b]$?