Databases Reference

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b.

Does the set of FDs {
C
→
D
,
B
→
C
} imply the JD
{
AB
,
BC
,
CD
}?

c.

Does the set of FDs {
A
→
B
,
B
→
C
} imply the JD
{
AB
,
BC
,
CD
}?

d.

Does the the JD
{
BC
,
ABD
} imply the JD
{
AB
,
BC
,
CD
}?

11.3 We know from Exercise 5.4 that the converse of Heath's Theorem is false. However, there's an extended

version of that theorem whose converse is true. Here it is:

Heath's Theorem
(
extended version
): Let relvar
R
have heading
H
and let
X
,
Y
, and
Z
be subsets of
H
such

that the union of
X
,
Y
, and
Z
is equal to
H
. Let
XY
denote the union of
X
and
Y
, and similarly for
XZ
. If
R
is

subject to the FD
X
→
Y
, then (a)
R
is subject to the JD
{
XY
,
XZ
}, and (b)
XZ
is a superkey for
R
.

Prove part (b) of this theorem. Prove also that (a) and (b) together imply that
X
→
Y
holds (the converse of the

extended theorem).

11.4

Consider the following JDs, both of which hold in relvar S:

{ { SNO , SNAME , CITY } , { CITY , STATUS } , { SNAME , CITY } }

{ { SNO , SNAME , CITY } , { CITY , STATUS , SNAME } }

I pointed out in the body of the chapter (in the section “Combining Components”) that although the first of these JDs

implied the second, decomposing relvar S on the basis of that second JD (even though it's irreducible) wouldn't be a

good idea. Why not?