1. Consider the relational schema R(A,B,C,D) and the following set of functional dependencies: S = {A -> B, BC -> A, D -> C}. A functional dependency X->Y (e.g., BD->A) is said to "logically follow" from S if when you compute the closure X+ ({B,D}+ in this case) it contains Y (A in this case). If X->Y does not logically follow from S, then there must be an instance of the relation R which satisfies the dependencies in S, but does not satisfy X->Y. (*Finding this may take a bit of thinking, and working backward.) For each of the functional dependencies X -> Y below show whether it logically follows from S or not. (So if the answer, is "yes", start by computing X+. If the answer is "no", show the appropriate (*smallest) instance.) (i) A -> C (ii) BD -> A (iii) CD -> A (iv) BCD -> A 2.Consider a relation R(A,B,C,D,E) satisfying the following dependencies: A->B, BC->E,ED->A. (a) List all keys for R. (b) Is R in 3NF? Justify your answer. (c) R is not in BCNF. Explain why? (d) Show the first step in a BCNF decomposition of R, including computing the projection of the dependencies onto the decomposed relations, and identifying keys for each subrelation. (e) Is the above decomposition lossless? Is it dependency-preserving? Are the subrelations in BCNF? 3NF? Explain each of your answers BRIEFLY. (f) Repeat step (d), if necessary, till you have decomposed R into BCNF relations. (g**) Argue whether there is a decomposition that preserves dependencies.