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GATE IT 2006 | Question: 60

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22 votes
22 votes

Consider a relation R with five attributes $V, W, X, Y,$ and $Z.$ The following functional dependencies hold:

$VY→ W, WX → Z,$ and $ZY → V.$

Which of the following is a candidate key for $R?$

  1. $VXZ$
  2. $VXY$
  3. $VWXY$
  4. $VWXYZ$
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2 Answers

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32 votes
32 votes

As we can see attributes  $X$ and $Y$ do not appear in the RHS of any FD and so they need to be part of any super/candidate key. So, candidate keys are: $VXY, WXY, ZXY$ as these three can determine any other attribute where as a proper subset of any of them cannot determine all other attributes.

$VXZ$ is not a super key as $Y$ is not there where as $VWXY$ and $VWXYZ$ are super keys but since their proper subsets are also super keys they are not candidate keys.

Answer is B.

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6 votes
6 votes
As we can see that w z and v can be generated from the given functional dependencies and there is no way we can generate X and Y therefore X and Y must be the part of candidate key: so, xy+ = xy & vxy + = vxywz So it turns out vxy is the key, ie. option (b) and so option c and d are super keys.
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