Database Modeling and Design

128 CHAPTER 6 Normalization
then the instance also contains the pair of rows obtained by interchanging
the Y values in the original pair. This includes situations
where only pairs of rows exist. Note that X and Y may contain either
single or composite attributes.
An MVD X ->> Y is trivial if Y is a subset of X, or if X union Y = RS.
Finally, an FD implies an MVD, which implies that a single row with a
given value of X is also an MVD, albeit a trivial form.
The following examples show where an MVD does and does not
exist in a table. In R1, the first four rows satisfy all conditions for the
MVDs X->>Y and X->>Z. Note that MVDs appear in pairs because of the
cross-product type of relationship between Y and Z=RS-Y as the two
right sides of the two MVDs. The fifth and sixth rows of R1 (when the X
value is 2) satisfy the row interchange conditions in the above definition.
In both rows, the Y value is 2, so the interchanging of Y values is
trivial. The seventh row (3,3,3) satisfies the definition trivially.
In table R2, however, the Y values in the fifth and sixth rows are different
(1 and 2), and interchanging the 1 and 2 values for Y results in a
row (2,2,2) that does not appear in the table. Thus, in R2 there is no
MVD between X and Y or between X and Z, even though the first four
rows satisfy the MVD definition. Note that for the MVD to exist, all rows
must satisfy the criterion for an MVD.
Table R3 contains the first three rows that do not satisfy the criterion
for an MVD, since changing Y from 1 to 2 in the second row results
in a row that does not appear in the table. Similarly, changing Z from 1
to 2 in the third row results in a nonappearing row. Thus, R3 does not
have any MVDs between X and Y or between X and Z.
R1: X Y Z R2: X Y Z R3: X Y Z
1 1 1 1 1 1 1 1 1
1 1 2 1 1 2 1 1 2
1 2 1 1 2 1 1 2 1
1 2 2 1 2 2 2 2 1
2 2 1 2 2 1 2 2 2
2 2 2 2 1 2
3 3 3