lossless and dependency preserving

Question

Consider a relation R(A, B, C, D, E, F, G) with set of functional dependencies
F = {AD → BF, CD → EGC, BD → F, E → D, F → C, D → F}
The relation R is decomposed into the following two relations after finding the minimal cover for the above set of functional dependencies R1(A, B, C, D, E) and R2(A, D, F, G).
Use the functional dependencies in the minimal cover to determine the nature of this decomposition.

(A) Decomposition is lossless and dependency preserving
(B) Decomposition is lossy and dependency preserving
(C) Decomposition is lossy and not dependency preserving
(D) Decomposition is lossless and not dependency preserving

Comments

I think option D is correct.

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Yes is it D?

Answer 1

Canonical cover of R is

AD>B, D>EG, E>D, F>C, D>F

It can be decomposed into two relations with the following functional dependencies.

R1: AD>B, D>E, E>D, D>C

R2:D>G, D>F

Between these two relations , F>C dependency is lost, hence it is not dependency preserving.

R1 and R2 have common attributes AD, whose closure using the above functional dependencies is A,D,F,G which is total of R2 hence it is lossless.