Union of Left-Total Relations is Left-Total
Jump to navigation
Jump to search
Theorem
Let $S_1, S_2, T_1, T_2$ be sets or classes.
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be left-total relations.
Then $\RR_1 \cup \RR_2$ is left-total.
Proof
Let both $\RR_1$ and $\RR_2$ be left-total.
Let $\RR = \RR_1 \cup \RR_2$.
Let $s \in S_1 \cup S_2$.
By the definition of union:
- $s \in S_1 \lor s \in S_2$
Thus $s \in S_i$ for $i \in \set {1, 2}$.
By definition of left-total relation, there is a $t \in T_i$ such that $\tuple {s, t} \in \RR_i$.
We have that $\RR$ is a superset of $\RR_i$.
Hence from Union is Smallest Superset:
- $\tuple {s, t} \in \RR_i \subseteq \RR \implies \tuple {s, t} \in \RR$
$\blacksquare$