Question

Consider the schema R (A, B, C, D) and the functional dependencies  A → B and C → D. If the decomposition is made as R₁ (A, B) and R₂ (C, D), then which of the following is TRUE ?
1. Preserves dependency but cannot perform lossless join
2. Preserves dependency and performs lossless join
3. Does not preserve dependency and cannot perform lossless join 
4. Does not preserve dependency but performs lossless join

Answer 1

Correct Answer - Option 1 : Preserves dependency but cannot perform lossless join

The correct answer is option 1.

Dependency preserving decomposition:

The Decomposition R with the dependency F into R₁ and R₂ with the dependency F₁ and F₂ is said to be dependency preserving if and only if F^l=( F1∪F2 )⁺. 

i.e every dependency of R we can determine using the decomposed relation.

R: Fl
R1 (A, B) : (F1)	R2 (C, D) : (F2)
A→B A→ A B→B	C→D C→C D→D

 

Here reflexive functional dependency removed because they obtain other attributes. So here Fl=( F1∪F2 )+  will be A→B and C→D will satisfy the F^l. So It Preserves dependency.

Lossless join Decomposition:

The Decomposition R into R₁ and R₂ is said to be lossless if R₁ ⋈ R₂ =R. 

Decomposition R into R₁ and R₂ is said to lossless if and only if the attribute common must be a key in either  R1 or  R2 or Both otherwise it lossy Decomposition or not lossless join.

The schema R (A, B, C, D) key is AC. R1 (A, B) and R2 (C, D) common attribute is empty so It not a key in  R1 or  R2 or Both relation. Hence It is not lossless join decomposition.
∴ Hence the correct answer is Preserves dependency but cannot perform the lossless join.