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Consider the following relation R(A, B, C, D, E, F, G) and set of functional dependencies.
F = {BCD → A, BC → E, A → F, F → G, C → D, A → G}
Which of the following is the minimal cover of F?
- a){BC → A, BC → E, A → F, C → D, A → G}
- b){BC → A, B → E, A → F, F → G, C → D}
- c){BC → A, BC → E, A → F, F → G, C → D}
- d){BC → A, BC → E, A → F, C → D}
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider the following relation R(A, B, C, D, E, F, G) and set of fun...
(1) RHS contain single attribute
A → F …..(1)
F → G …..(2)
A → G …….(3)
Hence FD' (3) is redundant.
Now consider
BCD → A
(BC)+ = BCEDAFG
It contains attribute D, D is redundant.
Minimal cover is {BC → A, BC → E, A → F, F → G, C → D}
So option (c) is correct.
Most Upvoted Answer
Consider the following relation R(A, B, C, D, E, F, G) and set of fun...
To find the minimal cover of a set of functional dependencies, we need to simplify the set by removing redundant or extraneous dependencies. Here is the step-by-step process to find the minimal cover of the given set of functional dependencies F:
1. Decompose the right-hand side of each dependency in F:
- BCD → A can be decomposed as BCD → A and BCD → C and BCD → D
- BC → E can be decomposed as BC → E
- A → F can be decomposed as A → F
- F → G can be decomposed as F → G
- C → D can be decomposed as C → D
- A → G can be decomposed as A → G
2. Remove extraneous attributes from the left-hand side of each dependency. If any attribute can be removed without changing the closure of the left-hand side, it is extraneous. Repeat this process until no more extraneous attributes can be removed.
- BCD → A remains the same
- BC → E remains the same
- A → F remains the same
- F → G remains the same
- C → D remains the same
- A → G remains the same
3. Remove redundant dependencies. If any dependency can be derived from the remaining dependencies, it is redundant and can be removed.
- BCD → A remains the same
- BC → E remains the same
- A → F remains the same
- F → G remains the same
- C → D remains the same
- A → G remains the same
4. Combine dependencies with the same left-hand side.
- BCD → A remains the same
- BC → E remains the same
- A → F remains the same
- F → G remains the same
- C → D remains the same
- A → G remains the same
5. The resulting set is the minimal cover of F:
- {BCD → A, BC → E, A → F, F → G, C → D, A → G}
Therefore, the correct answer is option C: {BCD → A, BC → E, A → F, F → G, C → D}.
1. Decompose the right-hand side of each dependency in F:
- BCD → A can be decomposed as BCD → A and BCD → C and BCD → D
- BC → E can be decomposed as BC → E
- A → F can be decomposed as A → F
- F → G can be decomposed as F → G
- C → D can be decomposed as C → D
- A → G can be decomposed as A → G
2. Remove extraneous attributes from the left-hand side of each dependency. If any attribute can be removed without changing the closure of the left-hand side, it is extraneous. Repeat this process until no more extraneous attributes can be removed.
- BCD → A remains the same
- BC → E remains the same
- A → F remains the same
- F → G remains the same
- C → D remains the same
- A → G remains the same
3. Remove redundant dependencies. If any dependency can be derived from the remaining dependencies, it is redundant and can be removed.
- BCD → A remains the same
- BC → E remains the same
- A → F remains the same
- F → G remains the same
- C → D remains the same
- A → G remains the same
4. Combine dependencies with the same left-hand side.
- BCD → A remains the same
- BC → E remains the same
- A → F remains the same
- F → G remains the same
- C → D remains the same
- A → G remains the same
5. The resulting set is the minimal cover of F:
- {BCD → A, BC → E, A → F, F → G, C → D, A → G}
Therefore, the correct answer is option C: {BCD → A, BC → E, A → F, F → G, C → D}.
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