Question 1 Suppose is a relation with attributes 12 . As a function of , tell how many superkeys has, if:

a) The only key is 1

Question 2 Consider a relation with schema (,,,) and FDs , , and .

a) What are all the nontrivial FDs that follow from the given FDs? You should restrict yourself to FDs with single attributes on the right side. [HINT: consider the closures of all nonempty sets of attributes}

b) What are all candidate keys of ?

Question 3 For a relation schema (,,) consider the following set of functional dependencies. ={,,,,,,,,}

1. Using the functional dependencies compute the minimal cover F. Give each step of your derivation with an explanation.

2. Show that there can be more than one minimal cover for a given set of functional dependencies. Give each step of your derivation with an explanation.

Question 4 Suppose we have relation (,,,,), with FDs ,,,, and we wish to project those FDs onto relation (,,). Give the functional dependencies that hold in S. Make sure that you give the minimal cover for functional dependencies in S. Show all the steps of your work.

Question 5 Show that the following rule hold by using the closure test we studied in the class. Make sure to show all steps of your work.

Pseudotransitivity. Suppose FDs 12 12 and 12 hold and the Bs are each among the Cs. Then 12 12 holds, where the Es are all those of the Cs that are not found among the Bs.