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

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Consider the binary decomposition

$R(A,B,C,D)\to R_1(A,C,D),R_2(B,C)$.

Which of the following statements are correct?

  1. If $C$ is a candidate key for $R_1$, the decomposition is lossless.
     
  2. For the decomposition to be lossless, tuples of $R_1$ and $R_2$ must always match one-to-one.
     
  3. If the decomposition is lossy, joining the projections can produce extra tuples not present in $R$.
     
  4. Even for a lossless decomposition, both projections can contain fewer tuples than the original relation.

1 Answer

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The common attributes are $R_1\cap R_2=\{C\}$.


Consider A.

If $C$ is a candidate key of $R_1$, then $C\to ACD$.

Thus, $(R_1\cap R_2)\to R_1$.

Therefore, the binary decomposition is lossless.

So A is correct.


Consider B.

A lossless join does not require a one-to-one correspondence between tuples.

One tuple in one component may correctly join with several tuples in the other component.

The important condition is that the join reconstructs exactly the original relation.

So B is incorrect.


Consider C.

A lossy decomposition can generate spurious tuples during the natural join.

Thus the join may contain tuples that were not in the original relation.

So C is correct.


Consider D.

Projection removes attributes and, under relational set semantics, duplicate projected tuples collapse into one tuple.

Therefore, a projected relation can have fewer tuples than the original relation even when the decomposition is lossless.

Hence D is correct.

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