Inverse Relation Equal iff Subset

Theorem
If a relation $$\mathcal{R}$$ is a subset of its inverse, then it equals its inverse:


 * $$\mathcal{R} \subseteq \mathcal{R}^{-1} \iff \mathcal{R} = \mathcal{R}^{-1} \iff \mathcal{R}^{-1} \subseteq \mathcal{R}$$

Proof

 * Suppose $$\mathcal{R} = \mathcal{R}^{-1}$$.

Then from the definition of set equality, $$\mathcal{R} \subseteq \mathcal{R}^{-1}$$ and $$\mathcal{R}^{-1} \subseteq \mathcal{R}$$.


 * Suppose $$\mathcal{R} \subseteq \mathcal{R}^{-1}$$.

Then:

$$ $$ $$ $$

Thus $$\mathcal{R} \subseteq \mathcal{R}^{-1} \implies \mathcal{R}^{-1} \subseteq \mathcal{R}$$ and it follows that $$\mathcal{R} = \mathcal{R}^{-1}$$.


 * Suppose $$\mathcal{R}^{-1} \subseteq \mathcal{R}$$.

Then:

$$ $$ $$ $$ $$

And the proof is complete.