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 either 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

The symmetric difference between two sets is also known as their:

disjoint union

Boolean sum.

However, both terms have different or more specialized meanings on $\mathsf{Pr} \infty \mathsf{fWiki}$, so will not be used here.

Another term seen occasionally is symmetric sum.

Some sources are pedantically explicit and use the term symmetric difference set.

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.

2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.): symmetric difference recognizes a further variant:

$S \mathop \nabla T$

Examples

Arbitrary Example $1$

Let:

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

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

Then their symmetric difference is given by:

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

Arbitrary Example $2$

Let:

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

$B = \set {3, 4, 5, 6}$

Then their symmetric difference is given by:

$A \symdif B = \set {1, 2, 5, 6}$

Also see

• Equivalence of Definitions of Symmetric Difference

• Results about symmetric difference can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): difference: 2. b. (of two sets)
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): symmetric difference
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): symmetric difference
• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Union
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): symmetric difference