Which of the following statements is TRUE ? D1 : The decomposition of the…

Answer: Only the decomposition of the 5-attribute schema is lossless; the decomposition of the 3-attribute schema is not always lossless.

Explanation for the first decomposition (R(A,B,C) → R1(A,B) and R2(A,C)):

• The intersection of the two subrelations is {A}.

• A decomposition is lossless only if the intersection functionally determines all attributes of at least one of the components. That would require A→B or A→C.

• No such functional dependency is given, so the decomposition is not guaranteed to be lossless.

Explanation for the second decomposition (R(A,B,C,D,E) with AD→B, C→DE, B→AE, AE→C, decomposed into R1(A,B,D) and R2(A,C,D,E)):

• The intersection of the two subrelations is {A,D}.

• Compute the closure of {A,D} under the given FDs to check if it determines all attributes:

• Start: AD+ = {A, D}.

• Using AD→B, add B: AD+ = {A, B, D}.

• Using B→AE, add E (and A is already present): AD+ = {A, B, D, E}.

• Using AE→C (A and E are present), add C: AD+ = {A, B, C, D, E}.

• Now AD+ contains all attributes of R, so {A,D} functionally determines the whole relation.

• Therefore the decomposition into R1(A,B,D) and R2(A,C,D,E) is lossless.

Conclusion: The first statement is false (not always lossless); the second statement is true (lossless).