Let R = ABCDE is a relational scheme with functional dependency set F = {A →…
2014
Let R = ABCDE is a relational scheme with functional dependency set F = {A → B, B → C, AC → D}. The attribute closures of A and E are
- A.
ABCD, φ
- B.
ABCD, E
- C.
Φ, φ
- D.
ABC, E
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Show answer & explanation
Correct answer: B
Solution: compute the attribute closures for A and E using the given functional dependencies.
Result: A+ = {A, B, C, D}; E+ = {E}.
Compute A+ (step-by-step):
Start with {A}.
Apply A → B to add B: {A, B}.
Apply B → C to add C: {A, B, C}.
Now A and C are present, so AC → D adds D: {A, B, C, D}.
Compute E+:
Start with {E}. There is no functional dependency with E on the left-hand side, so no new attributes can be added. Therefore E+ = {E}.
Why some answer choices are incorrect:
"ABCD, φ" is incorrect because E+ cannot be empty; closures always include the attribute(s) you start with, so E+ must include E.
"Φ, φ" is incorrect because neither closure can be empty; at minimum A+ contains A and E+ contains E, and in fact A+ expands to {A,B,C,D}.
"ABC, E" is incorrect because A+ should include D after applying AC → D once C is obtained.