GATE CSE 2017 | Set 1 | Question: 16

Question

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?

• A. $V \rightarrow W$
  $V \rightarrow X$
  $Y \rightarrow V$
  $Y \rightarrow Z$
• B. $V \rightarrow W$
  $W \rightarrow X$
  $Y \rightarrow V$
  $Y \rightarrow Z$
• C. $V \rightarrow W$
  $V \rightarrow X$
  $Y \rightarrow V$
  $Y \rightarrow X$
  $Y \rightarrow Z$
• D. $V \rightarrow W$
  $W \rightarrow X$
  $Y \rightarrow V$
  $Y \rightarrow X$
  $Y \rightarrow Z$

Comments

So when we say 

V ->W

and 

VW ->X

instead of that we can only say 

V -> X 

 

while also keeping in mind that  

W -> X

can't be concluded it like saying 

"only captains approval is required and coach approval and disapproval means nothing"

which is wronge (esp in football) 

---

v -> w  and  vw -> x   from these two w can be removed so left with  v -> w   and  v -> x
and  y -> vx   =>  y -> v   and   y -> x  

y -> v
v -> x 
y -> x
from  these 3   y -> x   can be eliminated
 

---

helpful to understand

Answer 1

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.

Comments

i don't think your 1st argument is correct  "dependency W−>X which is not implied by the question"  because then in option (a) v->x is present then it should not also be correct.

---

The problem is in reducing VW->X to W->X OR V->X.

From the original set of FD's we calculate V⁺ and W⁺

We get V⁺={VWX} and W⁺={W}.

If we consider option B

V→ W
W → X
Y → V
Y → Z

then we get W⁺={WX}

and if we consider option A

V→ W
V → X
Y → V
Y → Z

then W⁺={W} which is maintained. So we should go for option A.

---

yep

Answer 2

so, A is the ans.

Answer 3

COMMENT BELOW IF YOU HAVE ANY DOUBT