Jared - Thank you for the insights. My hesitation was caused by the fact that the definition of join doesn't seem to require that \\(C = A \vee B\\) must have three elements -- it seemed that I had the option of omitting * from C so that C would have only two (connected) elements. Now I see that if A, B, and C all have the same number of elements (i.e., the elements do not change), then the join is much easier to understand. I seem to have a "talent" for overly complicating solutions to relatively straightforward problems. I'll catch on ...

John - Thanks for the preview!