Definition:Symmetric Difference
Contents
Definition
The symmetric difference between two sets $S$ and $T$ is written $S * T$ and is defined as:
Definition 1
- $S * T := \left({S \setminus T}\right) \cup \left({T \setminus S}\right)$
Definition 2
- $S * T = \left({S \cup T}\right) \setminus \left({S \cap T}\right)$
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:
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
- Equivalence of Definitions of Symmetric Difference
- Results about symmetric difference can be found here.