Answer the following problems.

1. find out the closure (F+) for following set of the functional dependencies defined on R(a,b,c,d), where F – {c→a, ab→d, a→b, d→c}, i.e. what are all of the FDs implied by F.

2. Given the following set of functional dependencies {cf→bg, g→d, cdg→f, b→de, d→c} defined on R(b,c,d,e,f,g)

a. Is cf→e implied by FDs?

b. Is dg a superkey?

c. Is dg a candidate key?

d. Find a non-redundant cover.

e. Find a canonical cover.

f. What are the candidate keys?

3. Consider the following six relation schemes and their corresponding sets of functional dependencies. In each case identify

a. All the candidate keys

b. The highest normal form of the relation (2NF, 3NF, BCNF).

R1 (W, X, Y, Z) XY→Z, Y→ W

R2 (K,L,M,N) KL→N, K→M

R3 (P,Q,R,S) P→Q, Q→R, R→S

R4 (T,U,V) T→U, U→T, T→V

R5 (W,X,Y,Z) Z→W, YZ→W, WY→XZ

R6 (A,B,C,D) AB→D, AC→D

4. Determine a 3NF decomposition of the following relation scheme: University (Faculty, Dean, Department, Chair, Professor, Rank, Student}. University relation satisfies following set of functional dependencies:

{Faculty → Dean, Dean → Faculty, Department → Chair, Professor →Rank Chair, Department → Faculty, Student → DepartmentFacultyDean, ProfessorRank → DepartmentFaculty}

Is the decomposition non-loss and dependency preserving?

5. Consider the following relational scheme: R (a,b,c,d,e,f) and its corresponding set of FDs {d→a, be→c, ac→e, b→f, f→d, a→c}.

a. Is the following decomposition i) non-loss, ii) dependency preserving, iii) free of interrelational join constraints?

R1 (a,f,d), R2 (b,a,f), R3 (a,c,e)

b. Apply the BCNF decomposition on R.