Recall relvar CTXD from Chapter 9 (a sample value, repeated from Figure 9-3, is shown in Figure 12-2 overleaf). That relvar can be regarded as an extended version of relvar CTX as discussed earlier in the present chapter. The predicate is Teacher TNO spends DAYS days with textbook XNO on course CNO,^[116] and the sole key is {CNO,TNO,XNO}.

As we saw in Chapter 9, relvar CTXD suffers from redundancy;^[117] yet it’s in 5NF, which means no JDs (and therefore no MVDs, a fortiori) hold apart from trivial ones. In particular, therefore, the MVDs

     { CNO } →→ { TNO } | { XNO }

Figure 12-2. The 5NF relvar CTXD—sample value

do not hold^[118]—but they do hold in the projection of CTXD on {CNO,TNO,XNO}. For that reason, those MVDs are said to be embedded in the original relvar CTXD. In general, given some relvar R with heading H, an embedded dependency with respect to R is a dependency that doesn’t hold in R itself but does hold in the projection of R on some proper subset of H. As the example illustrates, therefore (and as was noted in Chapter 9, albeit in different words), embedded dependencies cause redundancy, but that redundancy can’t be eliminated by taking projections. Such redundancies thus correspond to constraints that must be separately stated and enforced (see Exercise 12.2).

Observe, incidentally, that the foregoing notion of embedding applies to JDs (and therefore to MVDs)^[119] but not to FDs. That is, given some relvar R and a projection of R whose heading includes both X and Y, the FD X → Y holds in that projection if and only if it holds in R itself. For example, the FD {CITY} → {STATUS} holds in relvar S as such and also in every projection of that relvar that retains both of those attributes.