Symmetric set - Alchetron, The Free Social Encyclopedia

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

S = S − 1

where S − 1 = { x − 1 : x ∈ S } . In other words, S is symmetric if x − 1 ∈ S whenever x ∈ 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 S = − S = { − x : x ∈ S } .

Examples

In R, examples of symmetric sets are intervals of the type ( − k , k ) with k > 0 , and the sets Z and { − 1 , 1 } .

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

If S is any subset of a group, then S S − 1 and S − 1 S are symmetric sets.

References

Symmetric set Wikipedia

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