PlanetMath: symmetric difference

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

symmetric difference

(Definition)

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

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

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

$\begin{pspicture}(0,0)(8,4) \begin{psclip} {\pscircle[fillstyle=vlines,hatchcolo... ...3.75){$A$} \rput(6.75,3.75){$B$} \rput(0,0){$.$} \rput(8,4){$.$} \end{pspicture}$

Properties

Suppose that

, and

are sets.

$A \triangle B=(A\setminus B) \cup (B\setminus A)$ .
$A \triangle B=A^c \triangle B^c$ , where the superscript denotes taking complements.
Note that for any set , 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
$\displaystyle \substack{n \\ \displaystyle{\triangle } \\ i=1} A_i.$
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:

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

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.

"symmetric difference" is owned by CWoo. [ full author list (3) | owner history (2) ]

(view preamble)

Other names:

set symmetric difference, symmetric set difference, symmetric difference between sets

Also defines:

symmetric difference operator

Keywords:

set, union, intersection

Attachments:: properties of symmetric difference (Derivation) by CWoo
symmetric difference on a finite number of sets (Derivation) by CWoo

Log in to rate this entry.
(view current ratings)

Cross-references: Boolean ring, fixed set, distributes over, intersection, power set, abelian group, group, properties, odd number, iff, multiple, associative, operation, commutative, satisfies, complements, superscript, discs, Venn diagram
There are 14 references to this entry.

This is version 16 of symmetric difference, born on 2001-11-16, modified 2008-05-01.
Object id is 916, canonical name is SymmetricDifference.
Accessed 22419 times total.

Classification:

AMS MSC:

03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

Pending Errata and Addenda

None.[ View all 9 ]

Discussion

forum policy

No messages.

Interact