Consider the relation 𝑋(𝑃,𝑄, 𝑅, 𝑆, 𝑇,π‘ˆ) with the following set of…

2015

Consider the relation 𝑋(𝑃,𝑄, 𝑅, 𝑆, 𝑇,π‘ˆ) with the following set of functional dependencies

𝐹 = {

Β Β Β Β Β Β Β Β {𝑃, 𝑅} β†’ {𝑆, 𝑇},

Β Β Β Β Β Β Β Β {𝑃, 𝑆,π‘ˆ} β†’ {𝑄, 𝑅}

}

Which of the following is the trivial functional dependency in 𝐹+, where 𝐹+ is closure of F ?

  1. A.

    {𝑃, 𝑅} β†’ {𝑆, 𝑇}

  2. B.

    {𝑃, 𝑅} β†’ {𝑅, 𝑇}

  3. C.

    {𝑃, 𝑆} β†’ {𝑆}

  4. D.

    {𝑃, 𝑆,π‘ˆ} β†’ {𝑄}

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Correct answer: C

Answer: {𝑃, 𝑆} β†’ {𝑆} is the trivial functional dependency.

Reason: a functional dependency is trivial when every attribute on the right-hand side also appears on the left-hand side.

  • Here, 𝑆 appears on both sides of {𝑃, 𝑆} β†’ {𝑆}, so it is trivial and therefore belongs to F+ without using the other dependencies.

  • Other choices either require deriving attributes from the given FDs (so they are non-trivial) or include right-hand attributes not present in the left-hand side, so they are not trivial.

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