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Consider $R(A,B,C,D,E,F,G,H,I,J)$ with $F=\{AB\to C,\ A\to DE,\ B\to F,\ F\to GH,\ D\to IJ\}$. The candidate key is $AB$.

Which of the following is a valid $\text{2NF}$ decomposition?

  1. $R_1(A,B,C)$
    $R_2(A,D,E,I,J)$
    $R_3(B,F,G,H)$
     
  2. $R_1(A,B,C,D,E)$
    $R_2(B,F,G,H)$
    $R_3(D,I,J)$
     
  3. $R_1(A,B,C,F,G,H)$
    $R_2(A,D,E,I,J)$
     
  4. Keep $R$ unchanged because $AB\to C$ uses the full key

1 Answer

1 1 vote

The candidate key is $AB$.

Consider the proper subset $A$.

We have $A\to DE$.

Also, $D\to IJ$.

$\therefore A^+=\{A,D,E,I,J\}$.

So all of $D,E,I,J$ are functionally determined by the proper subset $A$.

These non-prime attributes belong in an $A$-determined component $: R_2(A,D,E,I,J)$.

Now consider $B$.

$B\to F$ and $F\to GH$.

Thus, $B^+=\{B,F,G,H\}$.

Hence $F,G,H$ depend on proper subset $B$, so we obtain $R_3(B,F,G,H)$.

The attribute $C$ depends on $AB\to C$,

and neither $A$ nor $B$ alone determines $C$.

Therefore, $C$ is fully dependent on the composite key and stays with it $: R_1(A,B,C)$.

Notice that dependencies such as $D\to IJ$ and $F\to GH$ do not force further decomposition for $\text{2NF}$. 

Inside $R_2$, $A$ is a single-attribute key, and inside $R_3$, $B$ is a single-attribute key.

Thus all resulting relations are in $\text{2NF}$.

Therefore, the correct answer is A.

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