Definition:Symmetric Difference

Definition
The symmetric difference between two sets $S$ and $T$ is written $S * T$ and is defined as:
 * $S * T = \left({S \setminus T}\right) \cup \left({T \setminus S}\right)$

where:


 * $S \setminus T$ is the set difference between $S$ and $T$, defined as $S \setminus T = \left\{{x: x \in S \land x \notin T}\right\}$
 * $S \cup T$ is the union of $S$ and $T$, defined as $S \cup T = \left\{{x: x \in S \lor x \in T}\right\}$.

The symmetric difference can also be expressed as the set difference between their union and intersection:


 * $S * T = \left({S \cup T}\right) \setminus \left({S \cap T}\right)$

as is proved here.

Illustration by Venn Diagram
The red area in the following Venn diagram illustrates $S * T$:


 * VennDiagramSymmetricDifference.png

Also known as
The symmetric difference is also known as:
 * the disjoint union
 * the boolean sum.

Notation
There is no standard symbol for symmetric difference. The one used here: $*$ has been chosen somewhat arbitrarily; it's the one found by the author in the nearest work to hand.

The following are often found for $S * T$:


 * $S \oplus T$
 * $S + T$
 * $S \mathop \triangle T$

According to the following are also variants:


 * $S \mathop \Theta T$
 * $S \mathop \nabla T$