1. (5 pts) Consider the universal relation R = {A, B, C, D, E, F, G, H, I, J} and the set of functional dependencies F = {{A,B}{C}, {A}{D,E}, {B}{F}, {F}{G,H}, {D}{I,J}}. What is the key for R? Decompose R into 2NF and then 3NF relations.

2. (5 pts) Consider the relation R, which has attributes that hold schedules of courses and sections at a university. R = {CourseNo, SecNo, OfferingDept, CreditHours, CourseLevel, InstructorSSN, Semester, Year, Days_Hours, RoomNo, NoOfStudents}. Suppose that the following functional dependencies hold on R:
   1. {CourseNo} -> {OfferingDept, CreditHours, CourseLevel}
   2. {CourseNo, SecNo, Semester, Year} -> {Days_Hours, RoomNo, NoOfStudents, InstructorSSN}
   3. {RoomNo, Days_Hours, Semester, Year} -> {InstructorSSN, CourseNo, SecNo}

Try to determine which sets of attributes form keys of R. How would you normalize this relation to as high as 3NF? If there are multiple keys, choose one.

3. (5pt) Consider the relation R(MODEL#, YEAR, PRICE, MANUFACTURER, COLOR), which is abbreviated as R (M, Y, P, N, C), and the following set of F of functional dependencies: F={M -> N, {M,Y} -> P, N -> C}

1. Evaluate each of the following as a candidate key for R, giving reasons why it can or cannot be a key: {M}, {M,Y}, {M.C}
2. Based on the above key determination, state whether the relation R is in 3NF and in BCNF, giving proper reasons.
3. Consider the decomposition of R into D={R1(M,Y,P), R2(M,N,C)}. Is this decomposition lossless? Show why.
4. Consider the same decomposition in (c). Does this decomposition preserve dependencies? Show why.