Definition:Symmetric Difference/Definition 3
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Definition
The symmetric difference between two sets $S$ and $T$ is written $S \symdif T$ and is defined as:
- $S \symdif T = \paren {S \cap \overline T} \cup \paren {\overline S \cap T}$
where:
- $\cap$ denotes set intersection
- $\cup$ denotes set union
- $\overline S$ denotes the complement of $S$.
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
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets