## Symmetric difference

### Set

context | $ X,Y $ |

definiendum | $ X \triangle Y \equiv (X \smallsetminus Y) \cup (Y \smallsetminus X) $ |

### Discussion

Taken as 2-ary operation, $ X \triangle Y $ is commutative.

The symmetric difference is commutative, associative and distributive w.r.t. intersection.

The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.

#### Reference

Wikipedia: Symmetric difference