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2 votes
2 votes

Consider a relation schema $\mathrm{R}$ with attributes $(\mathrm{X}, \mathrm{Y}, \mathrm{Z}, \mathrm{U}, \mathrm{V}, \mathrm{W})$ and the functional dependency set $\mathrm{F}$ that holds on $\mathrm{R}:$
$$
\mathrm{F}=\{\mathrm{Y} \rightarrow \mathrm{U}, \mathrm{V} \rightarrow \mathrm{W}, \mathrm{U} \rightarrow \mathrm{V}, \mathrm{U} \rightarrow \mathrm{Y}, \mathrm{W} \rightarrow \mathrm{YU}\} \text {. }
$$
$\mathrm{R}$ is decomposed into two relations $\mathrm{P}(\mathrm{X}, \mathrm{Y}, \mathrm{U}, \mathrm{V}), \mathrm{S}(\mathrm{Y}, \mathrm{Z}, \mathrm{U}, \mathrm{W})$.
Which of the following is/are true regarding this decomposition?

  1. Decomposition is lossless.
  2. Decomposition is dependency preserving.
  3. $\text{P}$ is in $3 \mathrm{NF}$.
  4. $\mathrm{S}$ is in $3 \mathrm{NF}$.
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1 Answer

2 votes
2 votes
Decomposition is lossy because $\text{XYUV}$ intersection $\text{YZUW} =\mathrm{YU}$ which is NOT a key in either $\text{P}$ or $\text{S}.$

In the decomposition, dependencies are preserved i.e. union of FD set of $\mathrm{P}, \mathrm{S}$ is equivalent to $\mathrm{F}$.

In $\mathrm{P}$, there is no non-prime attribute, so it is in $3 \mathrm{NF}$.

In $\mathrm{S}$, there is no non-prime attribute, so it is in $3 \mathrm{NF}$.

3 Comments

How are the FDs of P and S getting preserved?

Because I am not able to derive the FD V → W ?
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Get V→Y from P and Y→ W from S. You get the result as V→W.
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How are you able to get Y → W from S ???
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