$A B \rightarrow C$ 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.