In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 ide ...
is said to be symmetric if it contains the
inverses of all of its elements.
Definition
In
set notation
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Defining ...
a subset
of a group
is called if whenever
then the inverse of
also belongs to
So if
is written multiplicatively then
is symmetric if and only if
where
If
is written additively then
is symmetric if and only if
where
If
is a subset of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
then
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 structure ...
structure of the vector space; that is, if
which happens if and only if
The of a subset
is the smallest symmetric set containing
and it is equal to
The largest symmetric set contained in
is
Sufficient conditions
Arbitrary
unions and
intersections of symmetric sets are symmetric.
Any
vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
in a vector space is a symmetric set.
Examples
In
examples of symmetric sets are intervals of the type
with
and the sets
and
If
is any subset of a group, then
and
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 whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
is symmetric.
See also
*
*
*
*
*
*
*
References
* R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
*
*
*
*
Group theory
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