symmetric set

Definition A subset $A$ of a group $G$ is said to be symmetric if $A=A^{-1}$, where $A^{-1}=\{a^{-1}:a\in A\}$. In other , $A$ is symmetric if $a^{-1}\in A$ whenever $a\in A$.

If $A$ is a subset of a vector space, then $A$ is said to be symmetric if it is symmetric with respect to the additive group structure of the vector space; that is, if $A=\{-a:a\in A\}$ [1].

0.0.1 Examples

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

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

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