Finding a relation in 3NF but not in BCNF
I've been reading many different sources on how to differentiate relations that are in 3NF/BCNF. And I've so far this is my understanding...
I will use this relation as an example...
R = {A, B, C, D, E}
and
F = {A -> B, B C - > E, E D -> A}.
Firstly we must find the keys of the relation. I used this video to help me do that. And I got
Keys = {ACD, BCD, CDE}
Now to make sure R is in BCNF, we must make sure that the left hand side of every functional dependency in F is one of the Keys. We instantly know this is not the case, because the first FD is A -> B and A is not one of the keys. So it is not in BCNF.
Now to make sure R is in 3NF, we must make sure that the left hand side of every functional dependency in F is one of the Keys OR the right hand side of every functional dependency in F is a subset of one of the Keys. If you look at the right hand side of every FD, they are B, E and A. These are each a subset of a Key, so this means that it is in 3NF.
So this is one of the rare cases (according to wiki) where a relation is in 3NF but not in BCNF. Is this method correct? Is it reliable? Am I missing anything?
Answer
First you need to learn superkeys, candidate keys, and primary attributes.
However, this rule of thumb helps:
A 3NF table that does not have multiple overlapping candidate keys is guaranteed to be in BCNF.
In other words, if the candidate keys in a 3NF relation are
- all atomic, or
- non-atomic but non-overlapping,
it is guaranteed that the relation is in BCNF.
The simplest relation which violates BCNF but meets 3NF has the following functional dependencies:
A,B -> C C -> B
In this case, candidate keys are (A,B) and (A,C).
It meets 3NF because
- the right-hand-side of all functional dependencies is a primary attribute.
It violates BCNF because
C -> B, but the left-hand-side is not a superkey.