1. Consider a relation R(A, B, C, D, E) with the following set of functional dependencies: F = {AB -> D, AC -> E, BC -> D, D -> A, E -> B}. Furthermore, R is decomposed into relation S(A, B, C) and other relations.

[a] Find all functional dependencies that hold in S.

[b] Determine all possible keys for S.

[c] Determine whether S is in BCNF and/or 3NF. Explain your answer.

2. Consider the relational schema R(City, Street, Zipcode), where a tuple (C, S, Z) is in R if and only if C has a building with street address at zip code Z in that city. Further, it is assumed that the nontrivial functional dependencies in R are:

F = {CS -> Z, Z -> C}.

[a] Determine all possible keys for R.

[b] Determine all possible S BCNF and 3NF violations.

[c] Decompose R as necessary into collection of relations that are in BCNF.

[d] If we decompose R as in [c], can the original relation be always recovered exactly by joining the tuples of the new ewlations in all possible ways? If not, explain why, giving an example to illustrate your answer