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.

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

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

Sufficient conditions
Arbitrary unions and intersections of symmetric sets are symmetric.

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

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

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

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