Some confusion with multiset index notation,

Question

I'm trying to use this multiset notation

$\Gamma= \{(\gamma_1,\mu_1), ... (\gamma_k, \mu_k)\}$

now, I know that a set (compared with a list) is inherently unordered.

So when I refer to elements in this set $\Gamma$ can I say

"...some pair $(\gamma_k, \mu_k)$ in $\Gamma$ ..."

or by using the index $k$, I am incorrectly only referring to the last pair in the set? Which wouldn't make sense, since a set is not ordered.

So is it correct to refer to an arbitrary pair in this set as $(\gamma_k, \mu_k)$ or use another index... such as $(\gamma_j, \mu_j)$ for $j \le k$?

Thanks,

Answer 1

By writing $\Gamma = \{(\gamma_1,\mu_1),\ldots,(\gamma_k,\mu_k)\}$ you have implicitly ordered the (multi)set $\Gamma$. This might not be part of the structure of $\Gamma$, but $(\gamma_k,\mu_k)$ certainly refers to the "last" element in the notation, so once you refer to that element using the letter $k$ you are using the implicit ordering.

In short, just use a different letter. You're probably not at a shortage.