Definition:Symmetric Difference

Definition

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

$S * T = \left({S \setminus T}\right) \cup \left({T \setminus S}\right)$

where:

• $S \setminus T$ is the set difference between $S$ and $T$, defined as $S \setminus T = \left\{{x: x \in S \land x \notin T}\right\}$
• $S \cup T$ is the union of $S$ and $T$, defined as $S \cup T = \left\{{x: x \in S \lor x \in T}\right\}$.

The symmetric difference can also be expressed as the set difference between their union and intersection:

$S * T = \left({S \cup T}\right) \setminus \left({S \cap T}\right)$

as is proved here.

Illustration by Venn Diagram

The red area in the following Venn diagram illustrates $S * T$:

Alternative Names

The symmetric difference is also known as:

the disjoint union

the boolean sum.

Notation

There is no standard symbol for symmetric difference. The one used here: $*$ has been chosen somewhat arbitrarily; it's the one found by the author in the nearest work to hand.

The following are often found for $S * T$:

• $S \oplus T$
• $S + T$
• $S \triangle T$

Also see

• Results about symmetric difference can be found here.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 5$: Complements and Powers
• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• J.A. Green: Sets and Groups (1965)... (previous)... (next): Exercise $1.7$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.4$, Exercise $7.1 \ \text{(a)}$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.2$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $6$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 8 \alpha$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$: Exercise $14$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $1.9$
• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.2$: Exercise $3$
• René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $\S 2$: Problem $2$