# Union of Right-Total Relations is Right-Total

## Theorem

Let $S_1, S_2, T_1, T_2$ be sets or classes.

Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be right-total relations.

Then $\mathcal R_1 \cup \mathcal R_2$ is right-total.

## Proof

Define the predicates $LT$ and $RT$ by:

• $LT\left(X\right) \iff X\text{ is left-total}$
• $RT\left(X\right) \iff X\text{ is right-total}$.

 $\displaystyle RT\left(\mathcal R_1\right) \land RT\left(\mathcal R_2\right)$ $\implies$ $\displaystyle LT\left(\mathcal R_1^{-1}\right) \land LT\left(\mathcal R_2^{-1}\right)$ Inverse of Right-Total is Left-Total $\displaystyle$ $\implies$ $\displaystyle LT\left(\mathcal R_1^{-1} \cup \mathcal R_2^{-1}\right)$ Union of Left-Total Relations is Left-Total $\displaystyle$ $\implies$ $\displaystyle LT\left(\left(\mathcal R_1 \cup \mathcal R_2\right)^{-1}\right)$ Union of Inverse is Inverse of Union $\displaystyle$ $\implies$ $\displaystyle RT\left(\mathcal R_1 \cup \mathcal R_2\right)$ Inverse of Right-Total is Left-Total

$\blacksquare$