You are here
Homesymmetric difference
Primary tabs
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.

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

$A\triangle B=A^{c}\triangle B^{c}$, where the superscript $c$ denotes taking complements.

Note that for any set $A$, the symmetric difference satisfies $A\triangle A=\emptyset$ and $A\triangle\emptyset=A$.

The symmetric difference operator is commutative since $A\triangle B=(A\setminus B)\cup(B\setminus A)=(B\setminus A)\cup(A\setminus B)% =B\triangle A$.

The symmetric difference operation is associative: $(A\triangle B)\triangle C=A\triangle(B\triangle C)$. This means that we may drop the parentheses without any ambiguity, and we can talk about the symmetric difference of multiple sets.

Let $A_{1},\ldots,A_{n}$ be sets. The symmetric difference of these sets is written
$\substack{n\\ \displaystyle{\triangle}\\ i=1}A_{i}.$ In general, an element will be in the symmetric difference of several sets iff it is in an odd number of the sets.
It is worth noting that these properties show that the symmetric difference operation can be used as a group law to define an abelian group on the power set of some fixed set.
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.
Mathematics Subject Classification
03E20 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections
Attached Articles
Corrections
set difference by matte ✓
typo by sidc ✓
Another Propertie by Bunder ✓
Venn diagram by CWoo ✓