# Inverse of Left-Total Relation is Right-Total

## Theorem

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

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

Then:

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

## Proof

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

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

Hence the result.

$\blacksquare$