Definition:Symmetric Difference

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Definition

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

Definition 1

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


Definition 2

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


Definition 3

$S * T = \left({S \cap \overline T}\right) \cup \left({\overline S \cap T}\right)$


Definition 4

$S * T = \left({S \cup T}\right) \cap \left({\overline S \cup \overline T}\right)$


Definition 5

$S * T := \left\{{x: x \in S \oplus x \in T}\right\}$


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.


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


Also known as

  • Disjoint union
  • Boolean sum


Notation

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

$S * T$

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