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Previous Year Question (PYQs)

An organization needs to maintain database having five attributes $A,B,C,D,E$. These attributes are functionally dependent on each other for which functional dependency set $F$ is given as:

$F={A\rightarrow BC,\ D\rightarrow E,\ BC\rightarrow D,\ A\rightarrow D}$

Consider a universal relation $R(A,B,C,D,E)$ with functional dependency set $F$. Also all attributes are simple and take atomic values only.

Assume that given table $R$ is decomposed in two tables:

$R_1(A,B,C)$ with functional dependency set

$f_1={A\rightarrow B,\ A\rightarrow C}$

and

$R_2(A,D,E)$ with FD set

$F_2={A\rightarrow D,\ D\rightarrow E}$

Which of the following option is true w.r.t. given decomposition?





Solution

For $R_1(A,B,C)$:

Given:

$A\rightarrow B$

$A\rightarrow C$

So,

$A^+={A,B,C}$

Therefore, $A$ is a key of $R_1$.

Since all dependencies have left side as key $A$, $R_1$ is in $3NF$.

Now for $R_2(A,D,E)$:

Given:

$A\rightarrow D$

$D\rightarrow E$

Here,

$A^+={A,D,E}$

So, $A$ is a key of $R_2$.

But dependency $D\rightarrow E$ violates $3NF$, because $D$ is not a super key and $E$ is a non-prime attribute.

Since the key $A$ is a single attribute, there is no partial dependency.

So, $R_2$ is in $2NF$ but not in $3NF$.


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