It can also be expressed as the union of two sets minus their intersection:

Or expressed by XOR operation:

In symmetric difference operation, an empty set is a unit element, and any element is its own inverse element.

To sum up, using symmetric difference operation, the power set of any set X is an Abelian group. Since all the elements in this group are negative elements of themselves, this group is actually a vector space on binary field Z2.

If X is finite, then the unit and the set of its elements form the basis of this vector space, and the dimension of the vector space is equal to the number of elements of X. This construction method can be used in graph theory to define the cycle space of a graph.

Extended data

Symmetric difference set: the symmetric difference set of set A and set B is defined as the set of all elements in set A and set B that do not belong to A∩B, that is, A△B = {x | x ∈ A ∪ B, X? A∩B}, that is, a△B =(a∪B)-(a∪B).

That is, A△B =(A-B)∨( B-A) Obviously, the symmetric difference set operation satisfies the commutative law: A△B=B△A, and the symmetric difference set is also called symmetric difference.