Design Theory Activity - Monday October 24
Menu of three activities
1. Proof similar to upcoming challenge problem
Prove the difference rule for multivalued dependencies. Specifically, consider a relation R, and
let AA, BB, and CC be three sets of attributes in R. Prove that if AA –>> BB and AA –>> CC hold
for R, then AA –>> (BB – CC) also holds, where – is the standard difference of attribute sets.
For simplicity you may assume that AA does not intersect BB or CC.
Do not assume that CC is a subset of BB.
Don't forget to account for the remaining attributes in R: those that are not in AA, BB, or
CC. Call them DD.
Feel free to introduce additional names for sets of attributes. For example, you may find
it convenient to give names to (BB – CC), (BB CC), and (CC – BB).
Your proof should be based on the formal definition of MVDs, not on other rules. It should have
roughly the following form: "Suppose AA –>> BB and AA –>> CC hold. Then for all tuples t and
u there exists a tuple v such that ... [fill in] ... To prove that AA –>> (BB – CC) holds, we need to
prove that for all tuples t and u there exists a tuple v such that ... [fill in] … Therefore AA –>>
(BB – CC) holds.
2. Proof dissimilar from any challenge problem
Consider a relation R and a set F of functional dependencies for R. Show that in order to
determine whether R is in Boyce-Codd Normal Form, we only have to check whether any of the
FDs in F violate BCNF (i.e., don’t contain a key on the left-hand side); we don’t have to check
other FDs that follow from the FDs in F. Formally, prove: If R has a nontrivial FD AA BB that:
violates BCNF,
follows from the FDs in F, and
does not appear in F
then R must also have an FD CC DD that violates BCNF and does appear in F.
(over)
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