Definition:Symmetric Set

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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 $A$.