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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
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1 Answer

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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 $(D).$

Answer:

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