I've attempted Mathew Doty's puzzle and I've made the same mistake as Keith E. Peterson – I've plugged in John's formula from [Lecture 6](https://forum.azimuthproject.org/discussion/1901/lecture-6-chapter-1-computing-adjoints#latest).
However, if we assume a complete lattice, is the following reasoning correct?

$$\begin{eqnarray} r(x,y) &=& \bigvee \\{a \in A : \; \Delta(a) \leq_{A\times A} (x,y) \\} \\\\ &=& \bigvee \\{a \in A : \; (a,a) \leq_{A\times A} (x,y) \\} \\\\ &=& \bigvee \\{a \in A : \; a \leq_A x, a \leq_A y \\} \\\\ &=& x \vee y \end{eqnarray}$$

Edit: I think the last step is wrong: initially I thought that \$$x\$$ and \$$y\$$ are in the set \$$R = \\{a \in A : \; a \leq_A x, a \leq_A y \\} \$$, but that's not true. The set \$$R\$$ might contain one of them if there is a relation between \$$x\$$ and \$$y\$$ (either \$$x \le y\$$ or \$$y \le x\$$), but generally there isn't (the set is not totally ordered).