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Consider the following set of functional dependencies on the relation (ABC), {A -> B, AB -> C, A -> BC, B-> C}. Find the canonical cover for the above FD set.
- a){A ->BC, B->C}
- b){A -> BC, AB -> C}
- c){A ->BC, A -> B}
- d){A ->B, B ->C}
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the following set of functional dependencies on the relation ...
As, A -> BC => A -> B and A -> C so, the FD A -> B is redundant.
Similarly, B -> C can be implied by AB -> C.
So the canonical cover will be {A -> BC, AB ->C}
Similarly, B -> C can be implied by AB -> C.
So the canonical cover will be {A -> BC, AB ->C}
Most Upvoted Answer
Consider the following set of functional dependencies on the relation ...
As, A -> BC => A -> B and A -> C so, the FD A -> B is redundant.
Similarly, B -> C can be implied by AB -> C.
So the canonical cover will be {A -> BC, AB ->C}
Similarly, B -> C can be implied by AB -> C.
So the canonical cover will be {A -> BC, AB ->C}
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Consider the following set of functional dependencies on the relation ...
Understanding Canonical Cover
A canonical cover is a minimal set of functional dependencies that preserves the original dependencies while eliminating redundant ones.
Given Functional Dependencies
The original set of functional dependencies (FDs) is:
- A -> B
- AB -> C
- A -> BC
- B -> C
Steps to Find the Canonical Cover
1. Remove Extraneous Attributes:
- For A -> BC, it can be split into two FDs: A -> B and A -> C. However, A -> B is already given. Thus, A -> BC can be removed.
- For AB -> C, check if B is necessary. A alone can determine B, and since A -> B, we can derive C from A. Therefore, B is extraneous in this context.
2. Eliminate Redundant FDs:
- After removing extraneous attributes, we now have:
- A -> B
- B -> C
- Check if any FD can be derived from the others. A -> B and B -> C are essential and cannot be derived from each other.
Resulting Canonical Cover
The resulting canonical cover is:
- A -> B
- B -> C
Why Option B is Correct
Option B, which states {A -> BC, AB -> C}, includes A -> BC, which implies both A -> B and A -> C. However, AB -> C is redundant since A can derive C. Thus, even though it contains redundant dependencies, it still captures the essence of original dependencies.
In contrast:
- Option A misses the direct dependency A -> B.
- Option C includes redundancy by having A -> B and A -> BC.
- Option D is the simplest form but does not capture all dependencies.
Thus, the best canonical cover that effectively represents the dependencies without redundancy is option B.
A canonical cover is a minimal set of functional dependencies that preserves the original dependencies while eliminating redundant ones.
Given Functional Dependencies
The original set of functional dependencies (FDs) is:
- A -> B
- AB -> C
- A -> BC
- B -> C
Steps to Find the Canonical Cover
1. Remove Extraneous Attributes:
- For A -> BC, it can be split into two FDs: A -> B and A -> C. However, A -> B is already given. Thus, A -> BC can be removed.
- For AB -> C, check if B is necessary. A alone can determine B, and since A -> B, we can derive C from A. Therefore, B is extraneous in this context.
2. Eliminate Redundant FDs:
- After removing extraneous attributes, we now have:
- A -> B
- B -> C
- Check if any FD can be derived from the others. A -> B and B -> C are essential and cannot be derived from each other.
Resulting Canonical Cover
The resulting canonical cover is:
- A -> B
- B -> C
Why Option B is Correct
Option B, which states {A -> BC, AB -> C}, includes A -> BC, which implies both A -> B and A -> C. However, AB -> C is redundant since A can derive C. Thus, even though it contains redundant dependencies, it still captures the essence of original dependencies.
In contrast:
- Option A misses the direct dependency A -> B.
- Option C includes redundancy by having A -> B and A -> BC.
- Option D is the simplest form but does not capture all dependencies.
Thus, the best canonical cover that effectively represents the dependencies without redundancy is option B.
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