Equivalence of Definitions of Symmetric Relation

Definition 1 implies Definition 3
Let $\mathcal R$ be a relation which fulfils the condition:
 * $\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \in \mathcal R$

Then:

Hence $\mathcal R$ is symmetric by definition 3.

Definition 3 implies Definition 2
Let $\mathcal R$ be a relation which fulfils the condition:
 * $\mathcal R \subseteq \mathcal R^{-1}$

Then by Inverse Relation Equal iff Subset:
 * $\mathcal R = \mathcal R^{-1}$

Hence $\mathcal R$ is symmetric by definition 2.

Definition 2 implies Definition 1
Let $\mathcal R$ be a relation which fulfils the condition:
 * $\mathcal R^{-1} = \mathcal R$

Then:

Hence $\mathcal R$ is symmetric by definition 1.