Imagine I want to present the list $$ \{(0,2),(0,-2),(-1,1),(1,-1),(1,2),(-1,-2) \} $$ in a shorter manner, by using "$\pm$". What is the best way to do that?
Initially, I thought that using, for example, $$ \{ (0,\pm2),(\pm1,\mp1),(\pm1,\pm2) \} $$ would be enough, where using "$\mp$" would give away the fact that $(\pm1,\mp 1)$ represents only two pairs and therefore $(\pm1,\pm2)$ represents also two pairs. However, another interpretation might suggest that $(\pm1,\pm2)$ represents four pairs, all the possible combinations $\{(1,2),(-1,2),(1,-2),(-1,-2)\}$. So this notation generates ambiguity.
I think this can be solved by instead writing $$ \{ \pm(0,2),\pm(1,-1),\pm (1,2) \} $$ or even $$ \pm\{ (0,2),(1,-1),(1,2) \} $$ I wonder, however, if there is a general approach to this kind of notations. What is the best way of using the "$\pm$" notation and why?
$$(\pm1,\pm2)$$ is indeed ambiguous, as you don't know if the two signs are synchronous or not and the expression can be read as two or four pairs.
On the opposite,
$$(\pm1,\mp2)$$ would certainly be understood as two pairs, otherwise the $\mp$ would be illogical.
So
$$\pm(1,2)$$ and $$\pm(1,-1)$$ are better, and as you are describing a set all combinations are allowed.
Anyway, had you been describing, say, a sum, then the expression would still have been ambiguous, by not knowing if the three $\pm$ are synchronized (six or eight terms ?).
A notation with the operator before the braces, meaning an implied distribution over the elements, is quite unusual.