Given the following set of FDs on schema R (V,W,X,Y,Z) {Z→V, W→Y, XY→Z, V→WX} State whether the following decomposition are loss-less-join decompositions or not. R1=(V,W,X) , R2=(V,Y,Z) R1=(V,W,X), R2=(X,Y,Z)
Question
Given the following set of FDs on schema R (V,W,X,Y,Z) {Z→V, W→Y, XY→Z, V→WX} State whether the following decomposition are loss-less-join decompositions or not. R1=(V,W,X) , R2=(V,Y,Z) R1=(V,W,X), R2=(X,Y,Z)
Solution
To determine if a decomposition is lossless, we need to check if the intersection of the decomposed relations, when functionally closed, includes all attributes of at least one of the decomposed relations.
- For R1=(V,W,X) , R2=(V,Y,Z):
The intersection of R1 and R2 is {V}. The closure of {V} under the given set of functional dependencies is {V,W,X,Y}, which does not include all attributes of either R1 or R2. Therefore, this decomposition is not lossless.
- For R1=(V,W,X), R2=(X,Y,Z):
The intersection of R1 and R2 is {X}. The closure of {X} under the given set of functional dependencies is {X,Z,V,W,Y}, which includes all attributes of both R1 and R2. Therefore, this decomposition is lossless.
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