Definition:Symmetric Difference/Definition 2
Definition
The symmetric difference between two sets $S$ and $T$ is written $S \symdif T$ and is defined as:
- $S \symdif T = \paren {S \cup T} \setminus \paren {S \cap T}$
where:
- $\setminus$ denotes set difference
- $\cup$ denotes set union
- $\cap$ denotes set intersection.
That is, $S \symdif T$ is the set of elements that are in either $S$ or $T$, but not both.
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: 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
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $7$
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $6$
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events: Exercise $3$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): symmetric difference
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Operations on Sets
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): symmetric difference: $\text {(ii)}$