Suppose R is a relation schema and F is a set of functional dependencies on R.…
2008
Suppose R is a relation schema and F is a set of functional dependencies on R. Further, suppose R₁ and R₂ form a decomposition of R. Then the decomposition is a lossless join decomposition of R provided that:
- A.
R₁ ∩ R₂ → R₁ is in F⁺
- B.
R₁ ∩ R₂ → R₂ is in F⁺
- C.
both R₁ ∩ R₂ → R₁ and R₁ ∩ R₂ → R₂ functional dependencies are in F⁺
- D.
at least one from R₁ ∩ R₂ → R₁ and R₁ ∩ R₂ → R₂ is in F⁺
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Correct answer: D
For a decomposition of relation R into sub-relations R₁ and R₂ to be lossless, the intersection of their attributes (R₁ ∩ R₂) must functionally determine at least one of the sub-relations. Mathematically, either (R₁ ∩ R₂) → R₁ or (R₁ ∩ R₂) → R₂ must be in the closure of F (F⁺). This condition guarantees that joining R₁ and R₂ on their common attributes reconstructs the original relation without generating spurious tuples.
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