Question 2 (Functional Dependencies - Closure)
[37 points]
(a) [10 points] Consider the relation schema R(A,B,C,D,E,F) with the functional dependencies F =
{A→B,AB C,BC→DE,AC →E, DE→F} and G={A→BC,B→CDE,CE,D→BCE,E→
D}.
(1) [5 points] Determine if the two sets F and G are equivalent or not. Explain your answer in detail.
(2) [5 points] Determine if AB is a candidate key of R with respect to F.
(b) [6 points] Consider the relation schema R(A,B,C,D,E) with the functional dependencies F = {A →→
C,B→D,BD → E, C →E}. Which of the following sets of attributes functionally determine CD?
Show each step.
• BC
• BD
• AE
(c) [6 points] Consider the relation schema R(A,B,C,D) with the functional dependencies F = {AB →
C,CD,BC}. We state that the FD C → B cannot be logically implied by the given F.
(1) [3 points] Prove why C → B cannot be logically implied.
(2) [3 points] Provide an example for your answer. (For the example, you can provide a table that
has tuples with some values in it.)
(d) [6 points] Let R(A,B,C,D,E,F) be a relation schema, and let F = {AB→D, CE→F,AC→DE,BD→
CF} be a set of functional dependencies (FDs). Which of the following attribute sets is a candidate
key?
• BD
• AC
• AB
(e) [4 points] Consider the relation schema R(A,B,C,D,E,F) with the functional dependencies F = {A →
B, CD →E,B→D,CA}. By using the algorithm for calculating attribute closures provided in the
lecture slides, calculate the closure of the following attribute set. Show each step.
• CE
(f) [5 points] Let R(A,B,C) be a relation schema, and let F = {A → B} be a set of functional dependencies
(FDs). Write down all the functional dependencies of the closure F+ of F and count them. Use the
exponential algorithm from the lecture for calculating the closure of functional dependencies.