Symmetric set

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.

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 ${\displaystyle S\cap -S.}$

Arbitrary unions and intersections of symmetric sets are symmetric.

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

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.