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GO Classes DPP | GATE CS, DA | DBMS | BCNF Decomposition

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2 2 votes

Consider $R(A,B,C,D)$ with $F=\{A\to B,\ C\to D\}$. Which of the following is a correct BCNF decomposition of $R$?

  1. $R_1(A,B)$
    $R_2(C,D)$
    $R_3(A,C)$
     
  2. $R_1(A,B,C)$
    $R_2(C,D)$
     
  3. $R_1(A,C)$
    $R_2(B,D)$
     
  4. $R_1(A,B)$
    $R_2(A,C,D)$

1 Answer

1 1 vote

First determine a candidate key.

Neither $A$ nor $C$ can be derived from other attributes.

Using $A\to B$ and $C\to D$, we get

$(AC)^+=\{A,B,C,D\}$

$\therefore AC$ is the candidate key.

Now examine $A\to B$.

$A$ is not a superkey of $R$, so this FD violates BCNF.

Decompose using $A\to B:$

$R_1(A,B)$ and $R'(A,C,D)$.

$R_1$ is in BCNF because $A$ is its key.

Now inspect $R'(A,C,D)$.

The projected FD $C\to D$ still holds.

But $C$ is not a superkey of $ACD$, so this also violates BCNF.

Decompose again into $R_2(C,D)$ and $R_3(A,C)$.

In $R_2$, $C$ is a key.

In $R_3$, there is no non-trivial projected FD.

Therefore, all three relations are in BCNF $:\boxed{{AB,\ CD,\ AC}}$

Answer : A

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