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Suppose we have a relation $R(X, Y, Z, W)$ with the FD's:

$$X \to Y, Y \to Z, Z \to W$$

Which one of the following decompositions is not lossless (i.e., for some instance of $R$, the natural join of the decomposed relations is not equal to $R$?

  1. $R_{1}(X, Y), R_{2}(Y, Z), R_{3}(Z, W)$
  2. $R_{1}(X, Y), R_{2}(X, Z), R_{3}(X, W)$
  3. $R_{1}(X, W), R_{2}(Y, W), R_{3}(Z, W)$
  4. None of the above (that is they are all lossless)
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Answer is right .
in table R1 R2 and R3, only $W$ is common and it is not superkey for any table. Hence, C is not lossless (is lossy).
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All are lossless ,Use the Algorithm Given in Navathe to Solve these type of questions.

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