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Let R (A, B, C, D) be a relational schema with the following functional dependencies : 
A → B, B → C, C → D and D → B. The decomposition of R into (A, B), (B, C), (B, D)

  1. gives a lossless join, and is dependency preserving
  2. gives a lossless join, but is not dependency preserving
  3. does not give a lossless join, but is dependency preserving
  4. does not give a lossless join and is not dependency preserving
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In this can we apply transitivity rule

A->B B->C THEN A->C

B->C C->D THEN B->D

C->D D->B THEN C->B

NOW FROM DECOMPOSITION IT IS LOSSLESS AND ALL DEPENDENCIES ARE PRESERVED

BUT FOR C->D WE CAN DERIVE BY JOINING C->B AND B->D
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here C---->B and B --->D

SO C--->D also  implied

so DP perserving

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