In a schema with attributes A, B, C, D, and E, the following set of functional…
2024
In a schema with attributes A, B, C, D, and E, the following set of functional dependencies is given: A → B, A → C, CD → E, B → D, E → A.
Which of the following functional dependencies is NOT implied by the given set?
- A.
CD → AC
- B.
BD → CD
- C.
BC → CD
- D.
AC → BC
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Correct answer: B
Answer: BD → CD
To test whether a functional dependency is implied, compute the closure of its left-hand side.
CD⁺: Start with {C, D}. From CD → E, get E. From E → A, get A. From A → B, get B. So CD⁺ contains A and C; hence CD → AC is implied.
BD⁺: Start with {B, D}. From B → D, no new attribute is added because D is already present. There is no dependency that derives C from B or D. So C is not in BD⁺; hence BD → CD is not implied.
BC⁺: Start with {B, C}. From B → D, get D. Now C and D give E by CD → E, and E → A gives A. So BC⁺ contains C and D; hence BC → CD is implied.
AC⁺: Start with {A, C}. From A → B, get B, and C is already present. So AC⁺ contains B and C; hence AC → BC is implied.
Only BD → CD is not implied, so option B is correct.