Inverses of Right-Total and Left-Total Relations

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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:

Inverse of Right-Total Relation is Left-Total

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


Inverse of Left-Total Relation is Right-Total

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