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Given the truth table of a Binary Operation \$as follows: $$\begin{array}{|l|l|l|l|} \hline {} \text{X} & \text{Y }& \text{X\Y }\\ \hline \text{1} & \text{0 }& \text{1 }\\ \hline \text{1} & \text{1}& \text{1 }\\ \hline \text{0} & \text{1 }& \text{0 }\\ \hline \text{0} & \text{0 }& \text{1 }\\ \hline \end{array}$$ Identify the matching Boolean Expression. 1.$X \$┐ Y$
2. $┐ X \$ Y$3.$┐ X \$┐ Y$
4. none of the options

Given that,

$$\begin{array}{|c|c|c|} \hline X & Y & X\ Y \\\hline 1 & 0 & 1 \\\hline 1 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 & 0 & 1 \\\hline \end{array}$$

Here, $X\$ Y\equiv Y \rightarrow X \equiv \neg Y \vee X \equiv X \vee \neg Y$Now, we can check each options, 1.$X \$\neg Y \equiv \neg Y \rightarrow X \equiv \neg(\neg Y) \vee X\equiv Y \vee X \equiv X \vee Y$
2. $\neg X \$ Y \equiv Y \rightarrow \neg X \equiv \neg Y \vee \neg X \equiv \neg X \vee \neg Y$3.$\neg X \$\neg Y \equiv \neg Y \rightarrow \neg X \equiv \neg(\neg Y) \vee \neg X \equiv Y \vee \neg X \equiv \neg X \vee Y$

None of the option matches.

So, the correct answer is $(D).$

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