Suppose we have a collection of sets $\{S_i\}_{i\in I}$. (To make it simple, we can assume these sets lie in some "universe set" $S$ and the index set $I$ is possibly finite.)
The union of these sets is defined as $\bigcup_{i\in I} S_i = \{a\in S | \exists i \colon a\in S_i\}$.
The intersection of these sets is defined as $\bigcap_{i\in I} S_i = \{a\in S | \forall i \colon a\in S_i\}$.
Similarly we can define the set operation $\mathbf{\{a\in S | \exists! i \colon a\in S_i\}}$. I've seen it a lot of times, but does this have an actual name?
It generalizes the symmetric difference between two sets, but the similarity stops there.
Well, since symmetric difference of two sets is denoted by $A\triangle B,$ I suppose you could do something like $$\triangle_{i\in I}S_i.$$ I've never seen it before, but it seems natural enough.
That said, you should make sure to explicitly define it for your readers before using it extensively. It seems that $n$-ary symmetric difference is usually defined quite differently.
If you want to be very technical, you could instead show this as $$\bigcup_{i\in I}\left(S_i\smallsetminus\left(\bigcup_{j\in I\smallsetminus\{i\}}S_j\right)\right),$$ but that seems like a pain.
Added: It might be better to go with $$!_{i\in I}S_i,$$ instead. It doesn't conflict with factorial notation or $n$-ary symmetric difference, and it's brief. I'm afraid I've been unable to find anything about a name for such an operation, though.