# Definition:Symmetric Difference/Definition 4

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## Definition

The **symmetric difference** between two sets $S$ and $T$ is written $S * T$ and is defined as:

- $S * T = \paren {S \cup T}\cap \paren {\overline S \cup \overline T}$

where:

- $\cap$ denotes set intersection
- $\cup$ denotes set union
- $\overline S$ denotes the complement of $S$.

### 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.

### 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 see

- Results about
**symmetric difference**can be found here.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**symmetric difference**: $\text {(ii)}$