Question:
A relation schema \(R(A, B, C, D)\) with functional dependencies \(F = \{AB \rightarrow C,\ C \rightarrow D,\ D \rightarrow A\}\) is in which highest normal form?
कार्यात्मक निर्भरताओं \(F = \{AB \rightarrow C,\ C \rightarrow D,\ D \rightarrow A\}\) वाला संबंध स्कीमा \(R(A, B, C, D)\) किस उच्चतम सामान्य रूप (Normal Form) में है?
Options:
- 3NF (Third Normal Form)
- BCNF (Boyce-Codd Normal Form)
- 2NF only
- 1NF only
- 3NF (तृतीय सामान्य रूप)
- BCNF (बॉयस-कॉड सामान्य रूप)
- केवल 2NF
- केवल 1NF
Correct Answer: a)
Explanation:
1. Candidate keys are \(AB, BC, BD\).
2. Prime attributes are \(A, B, C, D\) (all attributes are prime!).
3. For \(AB \rightarrow C\): \(AB\) is a superkey (BCNF & 3NF satisfied).
4. For \(C \rightarrow D\): \(C\) is NOT a superkey, but \(D\) is a prime attribute! Thus 3NF is satisfied, but BCNF is violated.
5. Similarly for \(D \rightarrow A\): \(D\) is not superkey, but \(A\) is prime (3NF satisfied).
Therefore, the relation is in 3NF but not in BCNF.
2. Prime attributes are \(A, B, C, D\) (all attributes are prime!).
3. For \(AB \rightarrow C\): \(AB\) is a superkey (BCNF & 3NF satisfied).
4. For \(C \rightarrow D\): \(C\) is NOT a superkey, but \(D\) is a prime attribute! Thus 3NF is satisfied, but BCNF is violated.
5. Similarly for \(D \rightarrow A\): \(D\) is not superkey, but \(A\) is prime (3NF satisfied).
Therefore, the relation is in 3NF but not in BCNF.
उम्मीदवार कुंजियाँ \(AB, BC, BD\) हैं। सभी विशेषताएं (A, B, C, D) प्राइम हैं। निर्भरता \(C \rightarrow D\) में \(C\) सुपरकी नहीं है लेकिन \(D\) प्राइम विशेषता है, अतः यह 3NF की शर्त पूरी करता है परंतु BCNF में नहीं है।