Prove that every 3NF relation schema with just two attributes is also in BCNF. Prove that every schema that has at most one nontrivial FD is in BCNF.

Question

What will be an ideal response?

Answer 1

a. Suppose the schema R is in 3NF and that its attributes are AB. Then for every FD X ? A ? F ,either(1)A ? X , or (2) X is a superkey of R,or(3)A ? X , X is not a superkey, and A ? K ,whereK is a key of R. We will show that (3) is impossible in our situation, which implies that R is in BCNF.
Indeed, if (3) holds, then either {A} = K or X is empty and K = AB. The ?rst case is not possible because X ? A and A being a key implies that X is a superkey, contrary to the assumption that it is not (this assumption is part of (3)). The second case is impossible because {}?A implies that B ? A, contrary to the assumption that AB is a key (because it means that B alone is a key).

b. If R has exactly one non-trivial FD, X ? Y , X must be a superkey of R (if it is not, then by augmentation it is possible to make another non-trivial FD, violating the assumption). If R has no non-trivial FDs, the set of all attributes is its key and the schema is, again, in BCNF.