The definitions in this section parallel those given in earlier chapters for FDs and JDs and are therefore presented with little by way of further commentary.
Definition: Let H be a heading; then a multivalued dependency (MVD) with respect to H is an expression of the form X →→ Y, where X (the determinant) and Y (the dependant) are both subsets of H. Note: The phrase MVD with respect to H can be abbreviated to just MVD, if H is understood.
Note carefully that, like FDs and JDs, MVDs are defined with respect to some heading, not with respect to some relation or some relvar. Note too that from a formal point of view (again like FDs and JDs), MVDs are just expressions: expressions that, when interpreted with respect to some specific relation, become propositions that by definition can evaluate to either TRUE or FALSE.
Definition: Let relation r have heading H; let X →→ Y be an MVD, M say, with respect to H; and let Z be the attributes of H not contained in either X or Y. (In other words, Z is the complement with respect to H of the union of X and Y, and the union of X, Y, and Z is equal to H.) If r satisfies the JD
{XY,XZ}, then r satisfies M; otherwise r violates M.
Note that the foregoing definition is symmetric in Y and Z, whence it follows that r satisfies the MVD X →→ Y if and only if it satisfies the MVD X →→ Z (and we can therefore write them as a “one liner,” as noted in the previous section).
Definition: The MVD M holds in relvar R (equivalently, relvar R is subject to the MVD M) if and only if every relation that can ever be assigned to relvar R satisfies M. The MVDs that hold in relvar R are the MVDs of R.
From this definition and the previous one, it follows that R is subject to the MVD X →→ Y if and only if it’s subject to the MVD X →→ Z.
Fagin’s Theorem is the “stronger form of Heath’s Theorem” promised in Chapter 5. That is, where Heath’s Theorem gave only a sufficient condition for a relvar to be nonloss decomposable into two projections, Fagin’s Theorem gives both necessary and sufficient conditions. Of course, Fagin’s Theorem is “obvious,” given what we now know about JDs in general; with hindsight, there would never have been any formal need to define MVDs at all if JDs in general had been defined and properly investigated first. But Fagin’s Theorem was proved before JDs in general had been properly investigated, and it was a new and important result at the time; what’s more, it still has practical significance, inasmuch as MVDs do correspond to a fairly common kind of business rule, whereas the same can’t reasonably be said for “cyclic” n-way JDs for n > 2 as discussed in Chapter 9 and Chapter 10.