Example 1

Given: 
C -> BD
B -> AC

R(A, B, C, D) in BCNF?
C -> BD: is C a superkey?
Yes, since C+ = ABCD

B -> AC: is B a superkey?
Yes, since B+ = ABCD

Yes, R is in BCNF!


Example 2

Given: 
C -> BD
B -> A

R(A, B, C, D) in BCNF?

C -> BD: is C a superkey? 
Yes, since C+ = ABCD

B -> A: is B a superkey? 
No, since B+ = AB

R is NOT in BCNF

So, we decompose R into
R1 (B, A) primary key: B
R2 (C, D, B): primary key: C




Example 3: Find a canonical cover for the following set of functional dependencies.

C -> AD
AC -> B
B -> AC

1. Is A in the LHS of AC -> B extraneous? 
I.e., can we infer 'C -> B' from the above functional dependencies? 
Yes, since C+ = ABCD 

So,
C -> ABD
B -> AC

2. Is A in the RHS of 'C -> ABD' extraneous? 
I.e., can we infer 'C -> A' from {C -> BD, B -> AC}?

Yes.

So, 
C -> BD
B -> AC

3. Is B in the RHS of 'C -> BD' extraneous? 
I.e., can 'C -> B' be inferred from {C -> D, B -> AC}? 
No, since C+=CD.

4. Similarly, D in the RHS of 'C -> BD' is not extraneous.

5. Similarly, A in the RHS of 'B -> AC' is not extraneous.

6. Similarly, C in the RHS of 'B -> AC' is not extraneous

7. We stop.