# 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**.