Consider the attribute set ABCDEG and the FD set

AB → C, AC → B, AD → E, B → D, BC → A, E → G

Is the following decomposition of R(ABCDEG)

(a) dependency-preserving?

(b) lossless-join?

1: AB, BC, ABDE, EG

2: ABC, ACDE, ADG

Give proper justification for your answer.

1. Decomposition into AB, BC, ABDE, EG:

(a) It's not dependency preserving. The FDs not being preserved are AB → C, AC → B, BC → A

(b) It's not lossless join. Join between AB and BC is lossy as B ↛ AB and B ↛ BC (Decomposition into R1 and R2 is lossless join, only when R1∩R2 → R1 or R1∩R2 → R2)

2. Decomposition into ABC, ACDE, ADG:

(a) It's not dependency preserving. The FDs not being preserved are B → D, E → G

(b) It's lossless join. Join between ABC and ACDE is lossless as AC → ABC and then join of the result with ADG is again lossless as AD → ADE.

1. loosy and Non dependency preserving.

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