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{ S , M } { R }
{ M } { D , Y }
{ S } { B }
{ B } { Z , C }
{ Z , C } { H }
{S,M} is a key. The BCNF procedure yields {S,M,R}, {M,D,Y}, {S,B}, {B,Z,C}, and {Z,C,H}. No FDs
are lost.
CHAPTER 7
7.1
See the body of the chapter.
7.2 The closure F + of a set F of FDs is the set of all FDs implied by those in F . The closure of the set of FDs that
hold in the shipments relvar SP is given in the answer to Exercise 4.1.
7.3 The FD X Y is satisfied if and only if whenever two tuples agree on X , they also agree on Y (I'm
deliberately giving this definition in a pretty loose form). So:
If two tuples agree on X , they certainly agree on every subset Y of X , so the reflexivity rule is reasonable.
If two tuples agree on XZ , they certainly agree on Z . They also certainly agree on X and hence, if X Y is
satisfied, on Y as well; hence they agree on YZ , and so the augmentation rule is reasonable.
If two tuples agree on X and X Y is satisfied, they agree on Y . If they agree on Y and Y Z is satisfied,
they agree on Z . So the transitivity rule is reasonable.
7.4
See the body of the chapter.
7.5
Let U denote the intersection of Z and Y and let V denote the difference Z - Y between Z and Y (in that order).
Then:
1.
X Y (given)
2.
Z W (given)
3.
X U (1, decomposition)
4.
XV UV (3, augmentation)
5.
XV Z (simplifying 4)
6.
XV W (5, 2, transitivity)
7.
XV YW (1, 6, composition; this completes the proof)
The rules used in this proof are indicated in the comments. The following rules are all special cases of
Darwen's theorem: the union, transitivity, and augmentation rules. So too is the following useful rule:
If X Y and XY Z , then X Z .
Note: This latter is a special case of what's sometimes called the pseudotransitivity rule, which in its general form
looks like this:

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