R = (J, K,L, M,N, P)
and functional dependencies:
F={J→KLM, KL→MN, K→M, M→J}

5. a) Compute K+ (the closure of the attribute set {K}).

(b) Using the Armstrong’s axioms and the union, decomposition, and pseudotransitivity rulee, prove that JP is a superkey (Hint: the proof goes step-by-step, and each step you will apply one of the 6 rules. The closure of JP will do the job but that’s NOT the answer we are asking for).

(c) Find a canonical cover of F; give each step of your derivation with an explanation. Once you reach your canonical cover, explain why you can’t further simplify the set of FDs.

(d) Give a 3NF decomposition of R based on the canonical cover you find.

6. (a) Give a BCNF decomposition of R. At each decomposition step, you should specifically point out the violating FD in F+ that leads to the decompostion step. If a violating FD is not in F, you need to prove it using the attribute set closure algorithm or the 6 rules in Question 5.

(b) Your BCNF decomposition may not be dependency preserving. If it is dependency preserving, explain why. If it is not, identify a FD that is not preserved and write a single SQL query to check if that FD is satisfied whenever the database is updated (Hint: a FD is not preserved does not mean that the data instance in the relations will violate that FD, but it is just harder to check if that FD is satisfied.).