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Which functional dependency holds in given relation $\mathrm{R}(\mathrm{A}, \mathrm{B}, \mathrm{C})$ and why?  
$$
\begin{array}{|c|c|c|}
\hline \mathbf{A} & \mathbf{B} & \mathbf{C} \\
\hline 7 & 1 & 8 \\
\hline 7 & 2 & 5 \\
\hline 7 & 3 & 5 \\
\hline 5 & 8 & 8 \\
\hline
\end{array}
$$

  1. $\mathrm{AB} \rightarrow \mathrm{C}$    $\&\&$    $\mathrm{C} \rightarrow \mathrm{B}$
     
  2. $\mathrm{BC} \rightarrow \mathrm{A}$    $\&\&$    $\mathrm{B} \rightarrow \mathrm{C}$
     
  3. $\mathrm{BC} \rightarrow \mathrm{A}$    $\&\&$    $\mathrm{A} \rightarrow \mathrm{C}$
     
  4. $\mathrm{AC} \rightarrow \mathrm{B}$    $\&\&$    $\mathrm{B} \rightarrow \mathrm{C}$

5 Answers

1 1 vote
$A B \rightarrow C$ satisfies condition1 (all the values of $A B$ are unique), but $C \rightarrow B$ does not satisfy any of the three conditions ( C has not unique values, B has not all same values and the value of C is repeated two times $(8,5)$ then the value of $B$ is not repeated), so option $A$ is incorrect.
$\mathrm{BC} \rightarrow \mathrm{A}$ satisfies condition 1, and $\mathrm{B} \rightarrow \mathrm{C}$ also satisfies condition 1, so option $B$ is correct.
$\mathrm{BC} \rightarrow \mathrm{A}$ satisfies condition 1, but $\mathrm{A} \rightarrow \mathrm{C}$ does not satisfy any of the three conditions, so option $C$ is incorrect.
$\mathrm{AC} \rightarrow \mathrm{B}$ does not satisfy any of the three conditions, and $\mathrm{B} \rightarrow \mathrm{C}$ satisfies condition 1 , so option $D$ is incorrect.

Since $\mathrm{BC} \rightarrow \mathrm{A} \& \& \mathrm{~B} \rightarrow \mathrm{C}$ holds in above relation, option $B$ will be the answer.
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