Definition:Symmetric Difference/Definition 5
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:
- $S \symdif T := \set {x: x \in S \oplus x \in T}$
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:
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: 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
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson