Definition:Symmetric Difference

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Definition

The symmetric difference between two sets $S$ and $T$ is written $S \symdif T$ and is defined as:

Definition 1

$S \symdif T := \paren {S \setminus T} \cup \paren {T \setminus S}$


Definition 2

$S \symdif T = \paren {S \cup T} \setminus \paren {S \cap T}$


Definition 3

$S \symdif T = \paren {S \cap \overline T} \cup \paren {\overline S \cap T}$


Definition 4

$S \symdif T = \paren {S \cup T}\cap \paren {\overline S \cup \overline T}$


Definition 5

$S \symdif T := \set {x: x \in S \oplus x \in T}$


where:

$\setminus$ denotes set difference
$\cup$ denotes set union
$\cap$ denotes set intersection
$\overline S$ denotes the complement of $S$
$\oplus$ denotes the exclusive or connective.


That is, it is the set of all elements of one of the two sets which are not also elements of the other set.


Illustration by Venn Diagram

The symmetric difference $S \symdif T$ of the two sets $S$ and $T$ is illustrated in the following Venn diagram by the red area:

VennDiagramSymmetricDifference.png


Also known as

  • Disjoint union
  • Boolean sum


Notation

There is no standard symbol for symmetric difference. The one used here, and in general on $\mathsf{Pr} \infty \mathsf{fWiki}$:

$S \symdif T$

is the one used in 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics.


The following are often found for $S \symdif T$:

$S * T$
$S \oplus T$
$S + T$
$S \mathop \triangle T$

According to 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics: Entry: symmetric difference:

$S \mathop \Theta T$
$S \mathop \triangledown T$

are also variants for denoting this concept.


Examples

Arbitrary Sets

Let:

$A = \set {1, 2, 3}$
$B = \set {2, 3, 4}$

Then their symmetric difference is given by:

$A \symdif B = \set {1, 4}$


Also see

  • Results about symmetric difference can be found here.


Sources