Given the relational schema ๐‘… = (๐‘ˆ,๐‘‰, ๐‘Š, ๐‘‹, ๐‘Œ, ๐‘) and the set ofโ€ฆ

2024

Given the relational schema ๐‘… = (๐‘ˆ,๐‘‰, ๐‘Š, ๐‘‹, ๐‘Œ, ๐‘) and the set of functional dependencies:

{๐‘ˆ โ†’ ๐‘‰, ๐‘ˆ โ†’ ๐‘Š, ๐‘Š๐‘‹ โ†’ ๐‘Œ, ๐‘Š๐‘‹ โ†’ ๐‘, ๐‘‰ โ†’ ๐‘‹}

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

  1. A.

    ๐‘‰๐‘Š โ†’ย ๐‘Œ๐‘

  2. B.

    ๐‘Š๐‘‹ โ†’ ๐‘Œ๐‘

  3. C.

    ๐‘‰๐‘Š โ†’ ๐‘ˆ

  4. D.

    ๐‘‰๐‘Š โ†’ ๐‘Œ

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Correct answer: A, B, D

Answer:

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

  • Derive WX โ†’ YZ:

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

  • Derive VW โ†’ YZ:

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

  • Derive VW โ†’ Y:

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

  • Why VW โ†’ U is not derivable:

    The given FDs include U โ†’ V and U โ†’ 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 โ†’ U cannot be derived.

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