7.17

Give an example of a relation schema $R^{\prime }$ and set $F^{\prime }$ of functional dependencies such that there are at least three distinct lossless decompositions of $R^{\prime }$ into BCNF.

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Given the relation $R^{\prime }=(A,B,C,D)$ the set of functional dependencies $F^{\prime }=A\to B,C\to D,B\to C$ allows three distinct BCNF decompositions.

$R_1=\{(A,B),(C,D),(B,C)\}$ is in BCNF as is $R_2=\{(A,B),(C,D),(A,C)\}$ $R_3=\{(B,C),(A,D),(A,B)\}$