# 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:

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

where $\oplus$ denotes the exclusive or connective.

### Illustration by Venn Diagram

The symmetric difference $S * T$ 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 $\mathsf{Pr} \infty \mathsf{fWiki}$:

- $S * T$

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

- 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