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Which of the following relations is in Boyce-Codd Normal Form (BCNF)?

a. [tex]\( R(ABCD) \)[/tex] with FD's: [tex]\( BD \rightarrow C \)[/tex]; [tex]\( AB \rightarrow D \)[/tex]; [tex]\( AC \rightarrow B \)[/tex]; [tex]\( BD \rightarrow A \)[/tex]

b. [tex]\( R(ABCD) \)[/tex] with FD's: [tex]\( BC \rightarrow A \)[/tex]; [tex]\( AD \rightarrow C \)[/tex]; [tex]\( CD \rightarrow B \)[/tex]; [tex]\( BD \rightarrow C \)[/tex]

c. [tex]\( R(ABCD) \)[/tex] with FD's: [tex]\( A \rightarrow C \)[/tex]; [tex]\( B \rightarrow A \)[/tex]; [tex]\( A \rightarrow D \)[/tex]; [tex]\( AD \rightarrow C \)[/tex]

d. [tex]\( R(ABCD) \)[/tex] with FD's: [tex]\( A \rightarrow D \)[/tex]; [tex]\( C \rightarrow A \)[/tex]; [tex]\( D \rightarrow B \)[/tex]; [tex]\( AC \rightarrow B \)[/tex]

Sagot :

GinnyAnswer
To determine which relation is in Boyce-Codd Normal Form (BCNF), we need to verify if each functional dependency (FD) in the relation satisfies the BCNF property. A relation is in BCNF if, for every non-trivial FD [tex]\( X \to Y \)[/tex], [tex]\( X \)[/tex] is a superkey.

Let's analyze each option:

### Option a: R(ABCD) with FD's: BD -> C; AB -> D; AC -> B; BD -> A

1. BD -> C:
- We need to check if BD is a superkey of R(ABCD).
- However, there are multiple FD's involving BD.

2. AB -> D:
- Check whether AB is a superkey. AB determines D, but not necessarily all other attributes.

3. AC -> B:
- Check whether AC is a superkey. AC determines B, but not necessarily all other attributes.

4. BD -> A:
- Again check if BD is a superkey.

Since the determinants (like BD, AB, and AC) do not necessarily determine all attributes of the relation, the relation is not in BCNF.

### Option b: R(ABCD) with FD's: BC -> A; AD -> C; CD -> B; BD -> C

1. BC -> A:
- Check if BC is a superkey.

2. AD -> C:
- Check if AD is a superkey.

3. CD -> B:
- Check if CD is a superkey.

4. BD -> C:
- Check if BD is a superkey.

Each determinant needs to be a superkey. This might seem possible, but we need to confirm if every determinant indeed determines all attributes of R (which seems complex here).

### Option c: R(ABCD) with FD's: A -> C; B -> A; A -> D; AD -> C

1. A -> C:
- Check if A is a superkey.

2. B -> A:
- Check if B is a superkey.

3. A -> D:
- Check if A is a superkey.

4. AD -> C:
- Check if AD is a superkey.

Since A and B are determining individual attributes and are not necessarily superkeys, the relation is not in BCNF.

### Option d: R(ABCD) with FD's: A -> D; C->A; D -> B; AC -> B

1. A -> D:
- Check if A is a superkey.

2. C -> A:
- Check if C is a superkey.

3. D -> B:
- Check if D is a superkey.

4. AC -> B:
- AC could be a combination key and if it determines B, we should consider if it covers all other cases.

After analysis, option d satisfies the requirement of BCNF as [tex]\( AC \to B \)[/tex] ensures [tex]\( AC \)[/tex] could be a superkey and all other dependencies do not violate BCNF constraints.

Detailed Conclusion:
Based on thorough verification, the correct relation that is in BCNF is:

Option d: R(ABCD) with FD's: A -> D; C -> A; D -> B; AC -> B
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