Given the following relation instance.

$$\begin{array}{|l|l|}\hline \text{X} & \text{Y} & \text{Z} \\\hline \text{1} & \text{4} & \text{2} \\ \text{1} & \text{5} & \text{3} \\ \text{1} & \text{6} & \text{3} \\ \text{3} & \text{2} & \text{2} \\\hline \end{array}$$

Which of the following functional dependencies are satisfied by the instance?

- $XY \rightarrow Z$ and $Z \rightarrow Y$
- $YZ \rightarrow X$ and $Y \rightarrow Z$
- $YZ \rightarrow X$ and $X \rightarrow Z$
- $XZ \rightarrow Y$ and $Y \rightarrow X$

## 5 Answers

**(b) is answer.**

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.

Since all $Y$ values are distinct FDs with $Y, YX$ and $YZ$ on LHS hold. So, option $B$ is correct.

In option A, $Z \to Y$ is violated as for same $Z$ value we have different $Y$ values.

Similarly in C, $X \to Z$ is violated and in D, $XZ \to Y$ is violated.