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Suppose we are told that R(A,B,C,D) is in BCNF, and that three out of the four functional dependencies (A)-(D) listed below hold for R.

Choose the Functional dependency that doesn't satisfy?

(A) A\rightarrowBCD

(B) BC\rightarrowA

(C) CD\rightarrowB

(D) D\rightarrowC

How the answer given is D ??? Please expain.

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It is given that the relation $R(A, B, C, D)$ is in BCNF. And we know that one out of the given four fds does not hold. i.e. only 3 holds.

(A) $A \rightarrow BCD$

(B) $BC \rightarrow A$

(C) $CD \rightarrow B$

(D) $D \rightarrow C$

Let's start by option A, i.e. we assume that A does not hold, while B, C and D holds

So, the candidate key will be: $D$, since $D^+ = \{ A, B, C, D\} $

But the fd $BC \rightarrow D$ does not satisfy the BCNF requirement. So, the $R(A, B, C, D)$ with three fds (B), (C) and (D) is not in BCNF.

Looking at option B, i.e. except B all three others are fds that hold.

We have candidate key $A$. Here $D\rightarrow C$ violates BCNF.

Looking at option C.

candidate keys are: $A$, $BC$ and $BD$.

here, $D\rightarrow C$ violates BCNF.

We are left with last option D.

Candidate key will be $A$, $BC$, $CD$. And these satisfy the BCNF requirement.

So, we can conclude that fd of option (D) is the one that should not hold for $R(A, B, C, D)$ to be in BCNF.

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