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GO Classes DPP | GATE CS, DA | DBMS | Non-Binary Losslessness

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Consider $R(A,B,C,D,E,F,G)$ with $F=\{AC\to BD,\ BC\to E,\ BE\to DF,\ AG\to EB\}$ and decomposition $R_1(A,B,C,D),$ $R_2(A,B,C,E,G),$ $R_3(B,E,F),$ $R_4(A,E,G)$.

Which of the following successive-combining sequences correctly proves that the decomposition is lossless?

  1. Combine $R_1$ with $R_2$, then combine the result with $R_3$, then with $R_4$.
     
  2. Combine $R_1$ with $R_3$ first.
     
  3. Combine $R_1$ with $R_4$ first.
     
  4. Combine $R_3$ with $R_4$ first.

1 Answer

1 1 vote

Start with $R_1$ and $R_2$.

Their common attributes are

$R_1\cap R_2=\{A,B,C\}$.

We have $AC\to BD$.

Since $A$ and $C$ are contained in $ABC$,

$ABC\to D$.

$\therefore ABC\to ABCD=R_1$.

So $R_1$ and $R_2$ combine losslessly.

Their combined schema is

$R_{12}(A,B,C,D,E,G)$.

Now consider $R_3(B,E,F)$.

$R_{12}\cap R_3={B,E}$.

We have $BE\to DF$.

Hence, $BE\to F$.

$\therefore BE\to BEF=R_3$.

So the second combination is also lossless.

The new schema contains $A,B,C,D,E,F,G$, which is the entire original schema.

Finally, $R_4(A,E,G)$ is already contained in this combined schema. 

Its intersection with the combined schema is $R_4$ itself, so adding it is trivially lossless.

Therefore, sequence A certifies the complete decomposition as lossless.

Answer : A

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