A relation $R$ is said to be partial functional if for all $x$, if $y R x$ and $y' R x$, then $y=y'$.
The union of two relations $R$ and $S$ is defined as follows:
$$y (R \cup S) x \iff y R x \text{ or } y S x.$$
My question is what conditions must be placed on two partial functional relations to ensure that their union is again partial functional?
If $R$ and $S$ are two partial functional relations then to check that their union is partial functional we have four cases to check.
We have $y(R\cup S)x$ and $y'(R\cup S)x$, so the four cases are:
$1)$ $yRx$ and $y'Rx$, which implies $y=y'$ since $R$ is partial functional.
$2)$ $ySx$ and $y'Sx$, which also impleis $y=y'$ again since $S$ is partial funcitonal.
$3)$ $yRx$ and $y'Sx$, and
$4)$ $ySx$ and $y'Rx$.
What conditions on $R$ and $S$ do we need to ensure that cases $3)$ and $4)$ imply that $y=y'$? I have no idea where to start.
Both case 3 and 4 lead to the condition if $yRx$ and $y'Sx$, then $y=y'$. This just means that the corresponding partial functions must agree on the intersection of their respective domains. This conditions is both necessary and sufficient to check whether the union is again partial functional.