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.