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Symmetric set

## Summary

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.