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

  • $V \rightarrow W$
  • $VW \rightarrow X$
  • $Y \rightarrow VX$
  • $Y \rightarrow Z$

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

  1. $V \rightarrow W$
    $V \rightarrow X$
    $Y \rightarrow V$
    $Y \rightarrow Z$
  2. $V \rightarrow W$
    $W \rightarrow X$
    $Y \rightarrow V$
    $Y \rightarrow Z$
  3. $V \rightarrow W$
    $V \rightarrow X$
    $Y \rightarrow V$
    $Y \rightarrow X$
    $Y \rightarrow Z$
  4. $V \rightarrow W$
    $W \rightarrow X$
    $Y \rightarrow V$
    $Y \rightarrow X$
    $Y \rightarrow Z$
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In option $B$ and option $D$ there is a dependency $W\to X$ which is not implied by the question and hence they are definitely wrong.

Now in option $C)$ $Y\to X$ can be removed as it can be implied as $Y\to V$ and $V\to X$.

Hence, option (A) is correct.

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