Aspire Faculty ID #18614 · Topic: UGC NET Computer Science Dec 2022 Shift II (Paper II) · Just now
UGC NET Computer Science Dec 2022 Shift II (Paper II)

Which of the following statements is correct, according to the following instance of the relational schema R(X, Y, Z)?
 XY Z 
1   1a 
 11  a
 21 b 
 32 b 

Solution

Functional dependency $A \to B$ means:

If two rows have the same value of $A$, then they must have the same value of $B$.

Now check each dependency.

For $X \to Y$:

When $X=1$, $Y=1$ in both rows.
When $X=2$, $Y=1$.
When $X=3$, $Y=2$.

So, same $X$ always gives same $Y$.

Therefore, $X \to Y$ is true.

For $X \to Z$:

When $X=1$, $Z=a$ in both rows.
When $X=2$, $Z=b$.
When $X=3$, $Z=b$.

So, same $X$ always gives same $Z$.

Therefore, $X \to Z$ is true.

For $Z \to X$:

When $Z=b$, we have two different values of $X$:

$X=2$ and $X=3$

So, $Z \to X$ is false.

For $Y \to Z$:

When $Y=1$, we get:

$Z=a$ and $Z=b$

So, $Y \to Z$ is false.

For $Y \to X$:

When $Y=1$, we get:

$X=1$ and $X=2$

So, $Y \to X$ is false.

Hence, only $X \to Y$ and $X \to Z$ are true.

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