GATE | GATE-CS-2017 (Set 1) | Question 15

The following functional dependencies hold true for the relational schema R{V, W, X, Y, Z}:

V -> W
VW -> X
Y -> VX
Y -> Z

Which of the following is irreducible equivalent for this set of functional dependencies?

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(A) A
(B) B
(C) C
(D) D


Answer: (A)

Explanation: Given

V -> W
VW -> X
Y -> VX
Y -> Z

We need to find the minimal cover of these FDs

Option B. W->X can  not implied by the given FDs, so incorrect



Option C. Y->X can be implied from Y->V and V->X, hence redundant

Option D. W->X can  not implied by the given FDs, so incorrect

Option A. Minimal cover of dependencies that can be extracted out as

  1. V -> W
  2. V -> X
  3. Y -> V
  4. Y -> Z

Therefore, option A is most appropriate.

Alternate Solution

Irreducible equivalent functional dependency is minimal cover.
Given,

  {V → W, VW → X, Y → VX, Y → Z}

W extraneous from VW, since we have V → W,

= {V → W, V → X, Y → VX, Y → Z}

X extraneous from VX, since we have V → X,

=  {V → W, V → X, Y → V, Y → Z}

This explanation is contributed by Mithlesh Upadhyay


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