Specific scenario regarding BCNF decomposition

Question

Say I have a relation ABCD with FD's (A->D and AB -> ABCD)

Will a decomposed relation ABC be in BCNF? According to the second FD, AB form a key and is therefore in BCNF, but if you only look at the FD A -> D, is the relation no longer in BCNF then?

Answer 1

If you decompose a given relation schema (to which given dependencies apply), the next task is to determine, for each individual dependency in the original set :

(a) which (if any) of the new, decomposed, schemas does it apply to ?
(b) how has the decomposition affected the very definition of the FD ?

Question (a) applies to your original A->D dependency. Question (b) applies, sort of, to your original AB->ABCD dependency. I say "sort of" because that version is quite "overstated". Given that A->D was already a given, it could just as well just say AB->C.