symmetric difference
The symmetric difference^{} between two sets $A$ and $B$, written $A\mathrm{\u25b3}B$, is the set of all $x$ such that either $x\in A$ or $x\in B$ but not both. In other words,
$$A\mathrm{\u25b3}B:=(A\cup B)\setminus (A\cap B).$$ 
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.

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$A\mathrm{\u25b3}B=(A\setminus B)\cup (B\setminus A)$.

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$A\mathrm{\u25b3}B={A}^{c}\mathrm{\u25b3}{B}^{c}$, where the superscript $c$ denotes taking complements^{}.

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Note that for any set $A$, the symmetric difference satisfies $A\mathrm{\u25b3}A=\mathrm{\varnothing}$ and $A\mathrm{\u25b3}\mathrm{\varnothing}=A$.

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The symmetric difference operator is commutative^{} since $A\mathrm{\u25b3}B=(A\setminus B)\cup (B\setminus A)=(B\setminus A)\cup (A\setminus B)=B\mathrm{\u25b3}A$.

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The symmetric difference operation^{} is associative: $(A\mathrm{\u25b3}B)\mathrm{\u25b3}C=A\mathrm{\u25b3}(B\mathrm{\u25b3}C)$. This means that we may drop the parentheses without any ambiguity, and we can talk about the symmetric difference of multiple^{} sets.

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Let ${A}_{1},\mathrm{\dots},{A}_{n}$ be sets. The symmetric difference of these sets is written
$$\begin{array}{c}n\\ \mathrm{\u25b3}\\ i=1\end{array}{A}_{i}.$$ 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 set.
Finally, we note that intersection^{} distributes over the symmetric difference operator:
$$A\cap (B\mathrm{\u25b3}C)=(A\cap B)\mathrm{\u25b3}(A\cap C),$$ 
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  20130322 11:59:41 
Last modified on  20130322 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 