The GATE Grind

GATE 2017 CS – Question 26

Databases · Integrity Constraints and Normal Forms · 1 mark · Multiple choice

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

$V \rightarrow W$

$VW \rightarrow X$

$Y \rightarrow VX$

$Y \rightarrow Z$

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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Correct answer: (A) $V \rightarrow W$; $V \rightarrow X$; $Y \rightarrow V$; $Y \rightarrow Z$

Explanation

Because $V \rightarrow W$, the $W$ in $VW \rightarrow X$ is unnecessary, so that becomes $V \rightarrow X$. Splitting $Y \rightarrow VX$ gives $Y \rightarrow V$ and $Y \rightarrow X$. The dependency $Y \rightarrow X$ then follows from $Y \rightarrow V$ and $V \rightarrow X$, so it is redundant. What remains is $V \rightarrow W$, $V \rightarrow X$, $Y \rightarrow V$ and $Y \rightarrow Z$.