Equivalence of Definitions of Symmetric Relation

Definition 1 implies Definition 3
Let $\RR$ be a relation which fulfils the condition:
 * $\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$

Then:

Hence $\RR$ is symmetric by definition 3.

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

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

Hence $\RR$ is symmetric by definition 2.

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

Then:

Hence $\RR$ is symmetric by definition 1.