Definition:Symmetric Difference/Definition 1
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Definition
The symmetric difference between two sets $S$ and $T$ is written $S \symdif T$ and is defined as:
- $S \symdif T := \paren {S \setminus T} \cup \paren {T \setminus S}$
where:
- $\setminus$ denotes set difference
- $\cup$ denotes set union.
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
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.4$
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $1$ Set Theory: $1$. Sets and Functions: $1.2$: Operations on sets
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \alpha$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $14$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $9$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): symmetric difference
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 2$: Problem $2$
- 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