Functional Dependency Theory

Consider the following two relational schemas and sets of functional dependencies:

a. (10 points) R(A,B,C,D,E) with functional dependencies D → B, CE → A.

b. (10 points) S(A,B,C,D,E) with functional dependencies A → E, BC → A, DE → B.

For each of the above schemas, decompose it into BCNF. Show all of your work and explain, at each step, which dependency violations you are correcting. Make sure you describe each step in your decomposition steps. (10 points each)

A set of attributes X is called closed (with respect to a given set of functional dependencies) if X⁺ = X. Consider a relation with schema R(A,B,C,D) and an unknown set of functional dependencies. For each closed attribute set below, give a set of functional dependencies that is consistent with it.

c. (5 points) All subsets of {A,B,C,D} are closed.

d. (5 points) The only closed subsets of {A,B,C,D} are {} and {A,B,C,D}.

e. (5 points) The only closed subsets of {A,B,C,D} are {}, {B,C}, and {A,B,C,D}.