In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements.

Definition

In set notation a subset ${\displaystyle S}$ of a group ${\displaystyle G}$ is called symmetric if whenever ${\displaystyle s\in S}$ then the inverse of ${\displaystyle s}$ also belongs to ${\displaystyle S.}$ So if ${\displaystyle G}$ is written multiplicatively then ${\displaystyle S}$ is symmetric if and only if ${\displaystyle S=S^{-1}}$ where ${\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.}$ If ${\displaystyle G}$ is written additively then ${\displaystyle S}$ is symmetric if and only if ${\displaystyle S=-S}$ where ${\displaystyle -S:=\{-s:s\in S\}.}$

If ${\displaystyle S}$ is a subset of a vector space then ${\displaystyle S}$ is said to be a symmetric set if it is symmetric with respect to the additive group structure of the vector space; that is, if ${\displaystyle S=-S,}$ which happens if and only if ${\displaystyle -S\subseteq S.}$ The symmetric hull of a subset ${\displaystyle S}$ is the smallest symmetric set containing ${\displaystyle S,}$ and it is equal to ${\displaystyle S\cup -S.}$ The largest symmetric set contained in ${\displaystyle S}$ is ${\displaystyle S\cap -S.}$

Sufficient conditions

Arbitrary unions and intersections of symmetric sets are symmetric.

Any vector subspace in a vector space is a symmetric set.

Examples

In ${\displaystyle \mathbb {R} ,}$ examples of symmetric sets are intervals of the type ${\displaystyle (-k,k)}$ with ${\displaystyle k>0,}$ and the sets ${\displaystyle \mathbb {Z} }$ and ${\displaystyle (-1,1).}$

If ${\displaystyle S}$ is any subset of a group, then ${\displaystyle S\cup S^{-1}}$ and ${\displaystyle S\cap S^{-1}}$ are symmetric sets.

Any balanced subset of a real or complex vector space is symmetric.

See also

• Absolutely convex set – convex and balanced setPages displaying wikidata descriptions as a fallback
• Absorbing set – Set that can be "inflated" to reach any point
• Balanced function – Construct in functional analysisPages displaying short descriptions of redirect targets
• Balanced set – Construct in functional analysis
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set whose intersection with every line is a single line segment
• Minkowski functional – Function made from a set
• Star domain – Property of point sets in Euclidean spaces

References

• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

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