Inverse of Left-Total Relation is Right-Total

Theorem

Let $\RR \subseteq S \times T$ be a relation on $S \times T$.

Let $\RR^{-1} \subseteq T \times S$ be the inverse of $\RR$.

Then:

$\RR$ is left-total if and only if $\RR^{-1}$ is right-total.

Proof

From Inverse of Inverse Relation, the inverse of $\RR^{-1}$ is $\RR$.

From Inverse of Right-Total Relation is Left-Total:

$\RR^{-1}$ is right-total if and only if $\RR$ is left-total.

Hence the result.

$\blacksquare$