Equivalence of Definitions of Symmetric Relation

Theorem
A relation $\mathcal R$ is symmetric iff it equals its inverse:
 * $\mathcal R^{-1} = \mathcal R$

Proof

 * Suppose $\mathcal R \subseteq S \times S$ is symmetric.

Thus $\mathcal R^{-1} \subseteq \mathcal R$ and, from Inverse Relation Equals iff Subset, $\mathcal R^{-1} = \mathcal R$.


 * Now, suppose $\mathcal R^{-1} = \mathcal R$.

Thus:

... so $\mathcal R$ is symmetric by the definition of a Symmetric Relation.

Comment
Some sources use this definition:
 * $\left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\} = \mathcal R$

as the definition of a symmetric relation.