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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a nonempty subset of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
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^ where S^ := \left\. If G is written additively then S is symmetric if and only if S = - S where - S := \. If S is a subset of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
then S is said to be a if it is symmetric with respect to the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structu ...
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 Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
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^ and S \cap S^ are symmetric sets. Any balanced subset of a real or complex
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
is symmetric.


See also

* * * * * * * *


References

* R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977. * * * * Group theory {{settheory-stub