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 Symmetric Relation.