$\begin{array}{|l|l|l|l|} \hline {} \text{X} & \text{Y }& \text{X\$Y }\\ \hline \text{1} & \text{0 }& \text{1 }\\ \hline \text{1} & \text{1}& \text{1 }\\ \hline \text{0} & \text{1 }& \text{0 }\\ \hline \text{0} & \text{0 }& \text{1 }\\ \hline \end{array}$
From the table, we can see that it is resembling the truth table of the implication operator for $Y\rightarrow X$.
or,
We can also get the same result by writing the expression for truth values 1.
The expression for $X\$ Y$ is:
$=$ $X\neg Y \vee XY \vee \neg X \neg Y$
On simplifying we get,
$=$ $\neg Y \vee X$
and, $\neg Y \vee X\equiv Y \rightarrow X$
therefore, $X \$ Y$ $=$ $Y\rightarrow X$
A. $X \$ \neg Y \equiv \neg Y\rightarrow X$ which is not same as the $Y\rightarrow X$.
B. $\neg X \$ Y \equiv Y\rightarrow \neg X$ which is not same as the $Y\rightarrow X$.
C. $\neg X \$ \neg Y \equiv \neg Y \rightarrow \neg X$ which is not same as the $Y\rightarrow X$.
D. None of the above
Therefore, D is the correct option.