# Definition:Symmetric Difference

## Contents

## Definition

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

### Definition 1

- $S * T := \paren {S \setminus T} \cup \paren {T \setminus S}$

### Definition 2

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

### 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.