Definition:Symmetric Difference/Definition 5

From ProofWiki
Jump to navigation Jump to search


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:

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

where $\oplus$ denotes the exclusive or connective.

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:



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.

Also see

  • Results about symmetric difference can be found here.