Definition:Symmetric Set
Definition
Let $G$ be a group.
Let $S \subseteq G$ such that:
- $\forall x \in S: x^{-1} \in S$
That is, for every element in $S$, its inverse is also in $S$.
Then $S$ is a symmetric subset of $G$, or (if $G$ is implicit) $S$ is a symmetric set.
Equivalently, $S \subseteq G$ is a symmetric set if and only if:
- $S = S^{-1}$
where $S^{-1}$ is the inverse of $S$.
Symmetric Set of Real Numbers
Let $\R$ be the set of real numbers.
Let $S \subseteq \R$ such that:
- $\forall x \in S: -x \in S$
That is, for every element in $S$, its negative is also in $S$.
Then $S$ is a symmetric subset of $\R$, or (if $\R$ is implicit) $S$ is a symmetric set.
Symmetric Subset of Vector Space
Let $V$ be a vector space over a subfield of $\C$.
Let $C \subseteq V$.
We say that $C$ is symmetric if whenever $x \in C$, we have $-x \in C$.
Symmetric Subset of Topological Group
Let $\struct {G, \odot, \tau}$ be a topological group.
Let $A \subseteq G$.
We say that $A$ is symmetric if and only if:
- $A = A^{-1}$
where $A^{-1}$ is the inverse of $A$.