1. Consider a schema R = {A,B,C,D,E} and the set F of FDs:

F = {AB --> CDE, A --> DE, D --> E}.

(i) Using the definition of 2NF, explain whether R is in 2NF or not.

(ii) Decompose R into 2NF, and not 3NF, preserving all dependencies.

2. Using the database schema {R_1,R_2,...,R_n} and the corresponding sets F_1,F_2,...,F_n of FDs obtained from question (1.ii):

(i) Using the definition of 3NF, explain whether each R_i is in 3NF or not.

(ii) Decompose the database schema into 3NF preserving all dependencies.

3. Consider a schema R = {A,B,C,D,E} and the set F of FDs:

F = {A --> BCDE, CD --> E, EC --> B}.

(i) Decompose R into 3NF preserving all dependencies. (Hint: notice that E is transitively upon the CK A, as is B.)

4. Consider a schema R = {A,B,C} and the set F of FDs:

F = {AB --> C, C --> B}.

(i) Using the definition of BCNF, explain whether or not R is in BCNF.

(ii) Decompose R into BCNF preserving all dependencies.

5. For this question we are only concerned with 4NF.

No MVD axioms are required in determining the answer.

Consider a schema R = {A,B,C,D,E,I} and the set MF of MVDs and FDs:

MF = {A -->--> BCD, B --> AC, C --> D}.

(i) Explain whether the MVD A -->--> BCD is trivial with respect to R.

(ii) Decompose R into 4NF. To test your understanding, I insist that you use the MVD A -->--> BCD first.

6. Prove that any relation scheme R in 3NF with respect to a set F of FDs must be in 2NF with respect to F.

(Hint: prove by contrapositive, namely, show that a partial dependency implies a transitive dependency.)

7. Either prove that any relation schema R in 4NF must be in BCNF or construct a counter-example to show this claim false.