**MD Puzzle 2':** Since,
$$
(a,b) \leq_{A\times A} (x,y) \\
\Longleftrightarrow \\
a \leq_A x \text{ and } b \leq_A y,
$$
it follows then that,
$$ r(x,y) = \bigvee \{a \in A : \; \Delta(a) \leq_{A\times A} (x,y) \} \\
\Longleftrightarrow \\
r(x,y) = \bigvee \{a \in A : \; a \leq_A x \text{ and } a \leq_A y \},$$
which is indeed a long form way to write,
$$ r(x,y) = x \wedge y .$$