# 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*: 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

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Sets