David - yes, that's cool. I should point out to everyone that the union of equivalence relations, the relation \\(\sim_P \cup \sim_Q\\), is the operation I'd been calling \\( \frown \\) in the lecture. The LaTeX name for this symbol is `\frown`, and I used that name because it's an unsatisfactory first attempt at defining \\(\sim_{P \vee Q}\\).