Given the relational schema 𝑅 = (π‘ˆ,𝑉, π‘Š, 𝑋, π‘Œ, 𝑍) and the set of…

2024

Given the relational schema 𝑅 = (π‘ˆ,𝑉, π‘Š, 𝑋, π‘Œ, 𝑍) and the set of functional dependencies:

{π‘ˆ β†’ 𝑉, π‘ˆ β†’ π‘Š, π‘Šπ‘‹ β†’ π‘Œ, π‘Šπ‘‹ β†’ 𝑍, 𝑉 β†’ 𝑋}

Which of the following functional dependencies can be derived from the above set?

Answer: A. π‘‰π‘Š β†’ π‘Œπ‘; B. π‘Šπ‘‹ β†’ π‘Œπ‘; D. π‘‰π‘Š β†’ π‘Œ β€” Answer: The following functional dependencies can be derived from the given set: VW β†’ YZ, WX β†’ YZ, and VW β†’ Y. Derive WX β†’ YZ: The given dependencies include…

  1. A.

    π‘‰π‘Š β†’Β π‘Œπ‘

  2. B.

    π‘Šπ‘‹ β†’ π‘Œπ‘

  3. C.

    π‘‰π‘Š β†’ π‘ˆ

  4. D.

    π‘‰π‘Š β†’ π‘Œ

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Correct answer: A, B, D

Answer:

The following functional dependencies can be derived from the given set: VW β†’ YZ, WX β†’ YZ, and VW β†’ Y.

  • Derive WX β†’ YZ:

    The given dependencies include WX β†’ Y and WX β†’ Z. By the union (composition) rule, these combine to give WX β†’ YZ.

  • Derive VW β†’ YZ:

    From V β†’ X, adding W gives VW β†’ WX. Since WX β†’ YZ (from above), transitivity yields VW β†’ YZ.

  • Derive VW β†’ Y:

    Either directly from VW β†’ YZ by projection, or by VW β†’ WX (via V β†’ X) together with WX β†’ Y.

  • Why VW β†’ U is not derivable:

    The given FDs include U β†’ V and U β†’ W, but there is no FD that produces U from V or W. There is no chain of dependencies that yields U in the closure of VW, so VW β†’ U cannot be derived.

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