Finding a relation in 3NF but not in BCNF

Question

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 1

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.

Answer 2

BCNF:

X->Y where Y can be prime or non-prime
Here,X must be a super key

3NF:

X->Y where Y is non-prime
then,
X must be a super key
else
X need not be super key

Hope,this helps