symmetric difference

The symmetric difference   between two sets $A$ and $B$, written $A\triangle B$, is the set of all $x$ such that either $x\in A$ or $x\in B$ but not both. In other words,

 $A\triangle B:=(A\cup B)\setminus(A\cap B).$

The Venn diagram  for the symmetric difference of two sets $A,B$, represented by the two discs, is illustrated below, in light red:

Properties

Suppose that $A$, $B$, and $C$ are sets.

Finally, we note that intersection  distributes over the symmetric difference operator:

 $A\cap(B\triangle C)=(A\cap B)\triangle(A\cap C),$

giving us that the power set of a given fixed set can be made into a Boolean ring  using symmetric difference as addition  , and intersection as multiplication.

 Title symmetric difference Canonical name SymmetricDifference Date of creation 2013-03-22 11:59:41 Last modified on 2013-03-22 11:59:41 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 20 Author CWoo (3771) Entry type Definition Classification msc 03E20 Synonym set symmetric difference Synonym symmetric set difference Synonym symmetric difference between sets Related topic SetDifference Related topic ProofOfTheAssociativityOfTheSymmetricDifferenceOperator Defines symmetric difference operator