GATE Overflow Test Series | Databases | Test 1 | Question: 15

Question

Given the following relation instance.
$$\begin{array}{|l|l|l|}\hline \text{X} & \text{Y} & \text{Z} \\
\hline \text{1} & \text{0} & \text{5} \\
\text{2} & \text{1} & \text{2} \\
\text{1} & \text{4} & \text{5} \\
\text{3} & \text{4} & \text{3} \\
\text{4} & \text{3} & \text{4} \\
\text{3} & \text{5} & \text{3} \\
\hline \end{array}$$

Which of the following functional dependencies are satisfied by the above relational instance $R(X,Y,Z)$? (Mark all the appropriate options)

• A. $XY \rightarrow Z$ and $Z \rightarrow Y$
• B. $YZ \rightarrow X$ and $Y \rightarrow Z$
• C. $YZ \rightarrow X$ and $X \rightarrow Z$
• D. $XZ \rightarrow Y$ and $Y \rightarrow X$

Answer 1

A functional dependency (FD) is a relationship between two sets of attributes.  For any relation $R,$ attribute set $Y$ is functionally dependent on attribute set $X$, if for every valid instance of $X,$ that value of $X$ uniquely determines the value of $Y.$ This relationship is indicated by the representation $:X \rightarrow Y.$

The left side of the above FD diagram is called the determinant, and the right side is the dependent.

If $A\to B$ then for each same value of $A$, $B$ value should be same. If all the $A$ values are distinct the FD hold irrespective of the $B$ values.

Thus in the given instance $XY \to Z, YZ \to X$ holds as all $XY$ and $YZ$ values are distinct.

But for single attribute determinant only $X \to Z$ holds.
    
 So, the correct answer is $C.$