I give the definition of symmetric difference of two sets in my article.
- The symmetric difference of two sets is the set of elements that are in either of the sets but not in their intersection.
A more formal definition:
- The symmetric difference of two sets $A$ and $B$ is $ A\,\triangle \,B=\left(A\setminus B\right)\cup \left(B\setminus A\right)$.
The definition of symmetry difference only considers two sets, so I feel somewhat uncomfortable when I want to say the sentence below.
The set $F$ is obtained by taking symmetric difference on the several sets $F_1, F_2, \cdots ,F_i$.
(That is to say: $F$ is obtained by $F_1 \triangle F_2 \triangle F_3 \cdots \triangle F_i$)
I don't know if there is any ambiguity or lack of rigor in this sentence. I would love to know a better expression of the above sentence.
PS: I always feel that I need to define the symmetric difference of multiple sets in my article, but it feels a little verbose.
The symmetric difference is associative and commutative. That means: $$(F_1\triangle F_2)\triangle F_3)=F_1\triangle(F_2\triangle F_3)=(F_1\triangle F_2)\triangle F_3$$And so on for more sets. You can efficiently define it as you would any iterate of an associative operation - repeat the operation $n$ times by applying successfully applying the operation on the output from the previous step. In the same way, if you have a definition for $a+b$, $a+b+c+d+e$ is defined to be the four-fold iteration of "$+$" and this is unambiguous as $(a+b)+c=a+(b+c)$, so the order is irrelevant.