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 {\displaystyle S} of a group G {\displaystyle G} is called symmetric if whenever s ∈ S {\displaystyle s\in S} then the inverse of s {\displaystyle s} also belongs to S . {\displaystyle S.} So if G {\displaystyle G} is written multiplicatively then S {\displaystyle S} is symmetric if and only if S = S − 1 {\displaystyle S=S^{-1}} where S − 1 := { s − 1 : s ∈ S } . {\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.} If G {\displaystyle G} is written additively then S {\displaystyle S} is symmetric if and only if S = − S {\displaystyle S=-S} where − S := { − s : s ∈ S } . {\displaystyle -S:=\{-s:s\in S\}.}
If S {\displaystyle S} is a subset of a vector space then S {\displaystyle S} is said to be a symmetric set if it is symmetric with respect to the additive group structure of the vector space; that is, if S = − S , {\displaystyle S=-S,} which happens if and only if − S ⊆ S . {\displaystyle -S\subseteq S.} The symmetric hull of a subset S {\displaystyle S} is the smallest symmetric set containing S , {\displaystyle S,} and it is equal to S ∪ − S . {\displaystyle S\cup -S.} The largest symmetric set contained in S {\displaystyle S} is S ∩ − S . {\displaystyle 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 , {\displaystyle \mathbb {R} ,} examples of symmetric sets are intervals of the type ( − k , k ) {\displaystyle (-k,k)} with k > 0 , {\displaystyle k>0,} and the sets Z {\displaystyle \mathbb {Z} } and ( − 1 , 1 ) . {\displaystyle (-1,1).}
If S {\displaystyle S} is any subset of a group, then S ∪ S − 1 {\displaystyle S\cup S^{-1}} and S ∩ S − 1 {\displaystyle S\cap S^{-1}} are symmetric sets.
Any balanced subset of a real or complex vector space is symmetric.
See also
- Absolutely convex set – Convex and balanced set
- Absorbing set – Set that can be "inflated" to reach any point
- Balanced function – Construct in functional analysisPages displaying short descriptions of redirect targets
- Balanced set – Construct in functional analysis
- Bounded set (topological vector space) – Generalization of boundedness
- Convex set – In geometry, set whose intersection with every line is a single line segment
- Minkowski functional – Function made from a set
- Star domain – Property of point sets in Euclidean spaces
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