symmetric difference

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

$$A\mathrm{△}B:=(A\cup B)\setminus (A\cap B).$$

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.

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

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

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

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

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

• Let $A_1,\mathrm{\dots },A_n$ be sets. The symmetric difference of these sets is written

  $$\begin{array}{c}n\\ \mathrm{△}\\ i=1\end{array}{A}_i.$$

  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:

$$A\cap (B\mathrm{△}C)=(A\cap B)\mathrm{△}(A\cap C),$$

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