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Consider the following set of functional dependencies F on the schema S(P, Q, R)
F = {P → QR, PQ → R, Q → R, P → Q}
The canonical cover of the above given set is
F = {P → QR, PQ → R, Q → R, P → Q}
The canonical cover of the above given set is
- a)P → QR and PQ → R
- b)P → QR and P → Q
- c)P → Q and Q → R
- d)P → QR and Q → R
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Consider the following set of functional dependencies F on the schema ...
P+ = {P, Q, R}
It covers, P → QR, PQ → R and P → Q
Also, Q → R
It is also minimal
Therefore, P → Q and Q → R is the canonical cover of the above given set.
It covers, P → QR, PQ → R and P → Q
Also, Q → R
It is also minimal
Therefore, P → Q and Q → R is the canonical cover of the above given set.
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Community Answer
Consider the following set of functional dependencies F on the schema ...
Understanding the Problem
The given functional dependencies on the schema S(P, Q, R) are:
- F = {P → QR, PQ → R, Q → R, P → Q}
We need to find the canonical cover for this set of functional dependencies.
What is a Canonical Cover?
A canonical cover is a minimal set of functional dependencies that preserves the closure of the original set. It should include:
- No redundant dependencies
- No extraneous attributes
Step-by-Step Analysis
- 1. Analyze Functional Dependencies
- P → QR implies that knowing P gives us both Q and R.
- PQ → R is redundant because if we know P, we already know R from P → QR.
- Q → R indicates that knowing Q is enough to determine R.
- P → Q shows that knowing P also gives us Q.
- 2. Remove Redundancies
- Since P → QR covers both Q and R, we can eliminate PQ → R.
- The dependency P → Q is also retained as it provides essential information.
- 3. Identify Extraneous Attributes
- The dependency Q → R does not depend on P, making it essential and non-redundant.
Final Canonical Cover
After evaluating the dependencies, the canonical cover can be summarized as:
- P → Q
- Q → R
These dependencies are minimal and cover the original set, confirming that option 'C' (P → Q and Q → R) is indeed the correct answer.
Conclusion
Thus, the canonical cover for the given functional dependencies is:
- P → Q
- Q → R
This ensures that all dependencies are necessary and sufficient for the closure of the original set.
The given functional dependencies on the schema S(P, Q, R) are:
- F = {P → QR, PQ → R, Q → R, P → Q}
We need to find the canonical cover for this set of functional dependencies.
What is a Canonical Cover?
A canonical cover is a minimal set of functional dependencies that preserves the closure of the original set. It should include:
- No redundant dependencies
- No extraneous attributes
Step-by-Step Analysis
- 1. Analyze Functional Dependencies
- P → QR implies that knowing P gives us both Q and R.
- PQ → R is redundant because if we know P, we already know R from P → QR.
- Q → R indicates that knowing Q is enough to determine R.
- P → Q shows that knowing P also gives us Q.
- 2. Remove Redundancies
- Since P → QR covers both Q and R, we can eliminate PQ → R.
- The dependency P → Q is also retained as it provides essential information.
- 3. Identify Extraneous Attributes
- The dependency Q → R does not depend on P, making it essential and non-redundant.
Final Canonical Cover
After evaluating the dependencies, the canonical cover can be summarized as:
- P → Q
- Q → R
These dependencies are minimal and cover the original set, confirming that option 'C' (P → Q and Q → R) is indeed the correct answer.
Conclusion
Thus, the canonical cover for the given functional dependencies is:
- P → Q
- Q → R
This ensures that all dependencies are necessary and sufficient for the closure of the original set.
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