# Definition:Symmetric Difference

## Definition

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

### Definition 1

$S \symdif T := \paren {S \setminus T} \cup \paren {T \setminus S}$

### Definition 2

$S \symdif T = \paren {S \cup T} \setminus \paren {S \cap T}$

### Definition 3

$S \symdif T = \paren {S \cap \overline T} \cup \paren {\overline S \cap T}$

### Definition 4

$S \symdif T = \paren {S \cup T}\cap \paren {\overline S \cup \overline T}$

### Definition 5

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

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.

That is, it is the set of all elements of one of the two sets which are not also elements of the other set.

## 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:

## Also known as

• Disjoint union
• Boolean sum

## Notation

There is no standard symbol for symmetric difference. The one used here, and in general on $\mathsf{Pr} \infty \mathsf{fWiki}$:

$S \symdif T$

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: symmetric difference:

$S \mathop \Theta T$
$S \mathop \triangledown T$

are also variants for denoting this concept.

## Examples

### Arbitrary Sets

Let:

$A = \set {1, 2, 3}$
$B = \set {2, 3, 4}$

Then their symmetric difference is given by:

$A \symdif B = \set {1, 4}$

## Also see

• Results about symmetric difference can be found here.