Definition:Symmetric Difference/Definition 2

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Definition

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

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

where:

$\setminus$ denotes set difference
$\cup$ denotes set union
$\cap$ denotes set intersection.


That is, $S * T$ is the set of elements that are in either $S$ or $T$, but not both.


Illustration by Venn Diagram

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

VennDiagramSymmetricDifference.png


Notation

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

$S * T$

is the one used in 1971: Allan Clark: Elements of Abstract Algebra.


The following are often found for $S * T$:

$S \oplus T$
$S + T$
$S \mathop \triangle T$ or $S \mathop \Delta T$

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.


Also see

  • Results about symmetric difference can be found here.


Sources