Symmetric set explained

In mathematics, a nonempty subset of a group 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.

References

• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.