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

Notice - irreducible equivalent

---

V --> W

VW --> X

implies

V --> X (by pseudo-transitivity rule)

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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.

From comments below.

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Everyone have a confusion on which one we need to take either V->X or W->X

For this take closure

V^+={V,W,X} it shows that V->W

So remove extraneous W and make

V->X

W^+={W}
In option B   and option D  there is a dependency W->X which is not implied by the question and hence they are definitely wrong

Then see this Y->V->X which indicate that Y->X so Y->X is redundent

 

Answer is

V->W

V->X

Y->V

Y->Z

Hence, option (A) is correct.

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

Is canonical cover equivalent to irreducible equivalent?

---

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.

Answer 2

so, A is the ans.