2.Consider the relation schema R = (N, Y, P, M, C) and assume that the following set of functional dependencies hold on R: F = { N ? M, NY ? P, M ? C} The letters can be interpreted as follows: R = (Model_Number, Year, Price, Manufacturing_Plant, Color).

a.[10 points] Evaluate each of the following as a candidate key for R, giving reasons why it can or cannot be a key: N, NY, NC.

b.[10 points] Find all the candidate keys of R.

c.[10 points] Give a lossless-join decomposition of R into Boyce-Codd normal form. Make sure to use the algorithm studied in class (slide 8.46) and to show all details.

d.[10 points] Does your decomposition preserve functional dependencies? Justify your answer.

e.[10 points] Is R in 3NF? Justify your answer.

f.[10 points] Show that the functional dependency NY ? P does not contain extraneous attributes.

g.[10 points] Show that F is already in canonical cover form.

h.[10 points] Use the algorithm we studied in class (slide 8.56) to find a lossless-join and dependency preserving decomposition of R into 3NF. Make sure to show all details.

i.[10 points] Consider the decomposition d = (R1, R2) where R1 = (N, Y, P) and R2 = (N, M, C). Is this decomposition lossless-join? Make sure to justify your answer and to show all details.