Let R (A1, A2, A3, A4) be a relation and F be a set of functional dependencies among attributes {A1,

A2, A3, A4} such that:

• {A1, A2, A3, A4}⁺ = {A1, A2, A3, A4}

• no proper, nonempty subset of {A1, A2, A3, A4} is equal to its closure with respect to the set F

of functional dependencies.

What is the set F of functional dependencies in R (compatible with the two conditions above) ?

Question 2:

Consider the following database schema: R(A, B, C, D, E, F, G, H) with the set of functional

dependencies { F-> A, AC-> E, E-> B, BG -> F, BE -> D, BDH -> E, D -> H, CD -> A, A E-> , AD -> BE}

2.1 Find the candidate keys of this schema. Show the full details of your work.

2.2 Find a BCNF decomposition of this schema (list both the relations and the corresponding set of

functional dependencies for each of the relations in the decomposition). 

2.3 Find a 3NF decomposition of this schema (list both the relations and the corresponding set of

functional dependencies).

Question 3:

Consider the schema with attributes ABCD and the following set of functional dependencies: { C -> AD,

AB -> C}. Produce a lossless BCNF decomposition for this schema. Is it dependency-preserving?

Explain why. If your BCNF decomposition is not dependency preserving, provide a dependency preserving 3NF decomposition.