12.1 (a) Relvar CTX in the body of the chapter is an example, of course, but it would be better if you could come up with an example from your own work environment. (b) Let C be a certain club, and let relvar R{A,B} be such that the tuple (a,b) appears in R if and only if a and b are both members of C. Then R is equal to the cartesian product of its projections R{A} and R{B}; thus, it’s subject to the JD {A,B} and, equivalently, to the following MVDs:

     { } ?? A | B

These MVDs aren’t trivial, since they certainly don’t hold in all binary relvars, and they’re not implied by a superkey either (the only key in R is the entire heading). It follows that R isn’t in 4NF. However, it’s certainly in BCNF, because it’s “all key.”