Editing Symmetric difference (section)

==''n''-ary symmetric difference==
Repeated symmetric difference is in a sense equivalent to an operation on a multitude of sets (possibly with multiple appearances of the same set) giving the set of elements which are in an odd number of sets.

The symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection:
<math display="block"> \Delta M = \left\{ a \in \bigcup M: \left|\{A \in M:a \in A\}\right| \text{ is odd}\right\}.</math>

Evidently, this is well-defined only when each element of the union <math display="inline">\bigcup M</math> is contributed by a finite number of elements of <math>M</math>.

Suppose <math>M = \left\{M_1, M_2, \ldots, M_n\right\}</math> is a [[multiset]] and <math>n \ge 2</math>. Then there is a formula for <math>| \Delta M|</math>, the number of elements in <math> \Delta M</math>, given solely in terms of intersections of elements of <math>M</math>:
<math display="block">| \Delta M| = \sum_{l=1}^n (-2)^{l-1} \sum_{1 \leq i_1 < i_2 < \ldots < i_l \leq n} \left|M_{i_1} \cap M_{i_2} \cap \ldots \cap M_{i_l}\right|.</math>