# Symmetric set

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In mathematics, a nonempty subset S of a group G is said to be symmetric if

${\displaystyle S=S^{-1}}$

where ${\displaystyle S^{-1}=\{x^{-1}:x\in S\}}$. In other words, S is symmetric if ${\displaystyle x^{-1}\in S}$ whenever ${\displaystyle x\in S}$.

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

## Examples

• In R, examples of symmetric sets are intervals of the type ${\displaystyle (-k,k)}$ with ${\displaystyle k>0}$, and the sets Z and ${\displaystyle \{-1,1\}}$.
• Any vector subspace in a vector space is a symmetric set.
• If S is any subset of a group, then ${\displaystyle S\cup S^{-1}}$ and ${\displaystyle S^{-1}\cap S}$ are symmetric sets.

## References

• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
• W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

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