As I mentioned near the beginning of this chapter, MVDs, unlike JDs in general, do have an axiomatization, or in other words a sound and complete set of rules for generating “new” MVDs from given ones. The rules in question are as follows:
If Y is a subset of X, then X →→ Y (“reflexivity”).
If X →→ Y and Z is a subset of W, then XW →→ YZ (“augmentation”).
If (a) the union of X, Y, and Z is equal to the pertinent heading H and (b) the intersection of Y and Z is a subset of X, then (c) X →→ Y | Z (“complementation”).
Now, these four rules aren’t nearly as easy to understand or remember as Armstrong’s rules are for FDs (or so it seems to me, at any rate). Partly for that reason, I won’t attempt to justify them here, nor will I show them in action. However, I will at least say that further rules can be derived from the original four, the following among them:
If X →→ Y and YZ →→ W, then XZ →→ W - YZ (“pseudotransitivity”).
If X →→ Y and X →→ Z, then X →→ YZ (“union”).
If X →→ YZ and W is the intersection of Y and Z, then X →→ Y - Z, X →→ Z - Y, and X →→ W (“decomposition”).
The following rules involve both MVDs and FDs:
If X → Y, then X →→ Y (“replication”).
If (a) X →→ Y, (b) Z → W, (c) W is a subset of Y, and (d) the intersection of Y and Z is empty, then (e) X → W (“coalescence”).
And the following is an additional derived rule:
If X →→ Y and XY → Z, then X → Z - Y (“mixed pseudotransitivity”).