Definition:Symmetric Difference/Definition 5

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Definition

Let $S$ and $T$ be any two sets.


The symmetric difference of $S$ and $T$ is the set which consists of all the elements which are contained in either $S$ or $T$ but not both:

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

where $\oplus$ 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


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


Sources