You won’t be surprised to hear there’s such a thing as a trivial MVD:

From this definition, it’s easy to prove the following theorem (see Exercise 12.7):

You probably won’t be surprised by the next definition, either:

As with FDs and JDs, “implied by keys” here could just as well be “implied by superkeys” without making any significant difference. Also, if M is trivial, it’s satisfied by every relation r with heading H, and so it’s satisfied by every relation r that satisfies R’s key constraints a fortiori. Thus, trivial MVDs are always “implied by keys,” trivially. So suppose M is nontrivial. Then it’s easy to prove the following theorem:

And now I can define 4NF:

However, given the various definitions and theorems already discussed in this section, we can see that the following “operational” definition is valid too:

Of course, if an MVD is implied by the keys of R, it certainly holds in R—i.e., it’s certainly “an MVD of R.” However, the converse is false: An MVD can hold in R without being implied by the keys of R (relvar CTX provides an example). Thus, the whole point about the 4NF definition is that the only MVDs that hold in a 4NF relvar are ones we can’t get rid of—which means ones implied by keys (including trivial ones as a special case).^[115]

Recall now from Chapter 10 the parallelism between the BCNF and 5NF definitions. In fact, that parallelism extends to the 4NF definition, too. That is, we have the following:

Now, in the BCNF and 4NF definitions, we can simplify “implied by the keys” to just “implied by some key”; as noted in Chapter 10, however, the same is not true for the 5NF definition. In that sense, 4NF resembles BCNF more than it does 5NF. On the other hand, 4NF also resembles 5NF more than it does BCNF, in the sense that the 4NF and 5NF definitions both rely on context—by which I mean that the MVDs and JDs that hold in a 4NF or 5NF relvar involve, at least implicitly, all of the attributes of that relvar, whereas the same is not true for BCNF. (As I said earlier, the point about MVDs always going in pairs is important. Nothing analogous applies to FDs.)

Recall now from Chapter 6 the concept of FD preservation. Essentially, the idea was as follows: If the FD X → Y holds in relvar R, then the recommendation is to decompose R—assuming that decomposition is desired at all, and assuming further that it’s done on the basis of some FD other than X → Y itself—in such a way that X and Y are kept together in the same projection. Well, the concept extends to MVDs too—that is, the recommendation still applies if we replace the FD X → Y by the MVD X →→ Y throughout.

In closing this section, let me state explicitly that: