Databases Reference
InDepth Information
If
X
→→
Y
and
Y
→→
Z
, then
X
→→
Z

Y
(“transitivity”).
3.
4.
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”).
5.
6.
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”).
7.
The following rules involve both MVDs and FDs:
8.
If
X
→
Y
, then
X
→→
Y
(“replication”).
9.
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”).
10.
EMBEDDED DEPENDENCIES
Recall relvar CTXD from Chapter 9 (a sample value, repeated from Fig. 9.3, is shown in Fig. 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
,
4
and the sole key is
{CNO,TNO,XNO}.
As we saw in Chapter 9, relvar CTXD suffers from redundancy;
5
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 }
4
This is the predicate I gave in Chapter 9, but a more accurate version might be:
Course CNO can be taught by teacher TNO and uses textbook
XNO, and teacher TNO spends DAYS days with textbook XNO on course CNO
. And we might want to add
DAYS is greater than zero
as well.
See Chapter 15 for further discussion.
5
As noted in Chapter 9, one of my reviewers disputed this claim. Again, see Chapter 15 for further discussion.