Is my application of Boyce-Codd NF correct in this exercise?

Question

EXAM revision, not homework.

I'm working on this exercise and I would like to check is my solution okay:

Given the following relationship, R, and set of functional dependencies, F: R = {A,B,C,D,E,F,G,H,I} F= { {C,D} -> {A}, {G} -> {E}, {C,D,E} -> {G,B,H}, {B} -> {F}, {H} -> {I} } decompose R into relations that satisfy Boyce-Codd normal form.

My answer: BOLD is primary key

(C,D,A)

(C,D,E,B,H)

(B,F)

(H,I)

(G,C,D)

(E,G)

Answer 1

I think the first four you derived are all in 5NF.

• {G CD}
• {EG}

But these two don't look right.

For one thing, one of your FDs is G->E. In a relation that contains only G and E, how did you determine that the right decomposition is

• {EG}

instead of

• {G E}

given that the FD G->E still has to hold?

At a glance, I don't see how you can derive G->CD from the FDs you gave.