BCNF，3NF和候选密钥

Question

I am having issues understanding how to determine if relations are in BCNF, 3NF, and in general identifying the candidate keys of a relation.

Consider relation R = (A, B, C, D) with the functional dependencies:

AB -> C
C -> D
D -> A

The questions included:

a. list the candidate keys of R
b. determine if R is in BCNF or 3NF.

The solution resolves

a. 3 candidate keys for R are AB, BC, and BD.
b. R is in 3NF, but not in BCNF.

I've read through What is the difference between 3NF and BCNF? and can understand the difference between 3NF and BCNF when referring to non-arbituary words within a database schema. I am ultimately lost when trying to determine the relationships with a reduced relation, as given in the problem.

Could someone explain how the candidate keys were determined above, and why R is in 3NF but not BCNF?

Answer 1

You are asking several questions at the same time.

1. How to compute the keys of a relation. In general, the simplest way is to compute the closure of every single combination of attributes and determine which ones are superkeys. From this set of superkeys, find those that are minimal: these are the candidate keys.

2. How to determine if a relation is BCNF.

FOr this you don't needs keys. All you need to do is test that every non-trivial functional dependency, the left hand side is a superkey of the relation.

How do you determine this for a FD? Simply compute the closure of the left hand side of the FD. If the result is all the attributes of the relation, then it is a superkey.

3. How to determine if the relation is 3NF?

FDs must be in Canonical form: one attribute in the right hand side.

For every non-trivial FD, either the left hand is a superkey (as in BCNF) or the right hand side is part of a key (any key).

This is a very good video that explains the differences:

BCNF vs 3NF