The negation of the Boolean expression
Question:

The negation of the Boolean expression $x \leftrightarrow \sim y$ is equivalent to:

  1. (1) $(x \wedge y) \vee(\sim x \wedge \sim y)$

  2. (2) $(x \wedge y) \wedge(\sim x \vee \sim y)$

  3. (3) $(x \wedge \sim y) \vee(\sim x \wedge y)$

  4. (4) $(\sim x \wedge y) \vee(\sim x \wedge \sim y)$


Correct Option: 1

Solution:

$p: x \leftrightarrow \sim y=(x \rightarrow \sim y) \wedge(\sim y \rightarrow x)$

$=(\sim x \vee \sim y) \wedge(y \vee x)$

$=\sim(x \wedge y) \wedge(x \vee y)$   $(\because \sim(x \wedge y)=\sim x \vee \sim y)$

Negation of $p$ is

$\sim p=(x \wedge y) \vee \sim(x \vee y)=(x \wedge y) \vee(\sim x \wedge \sim y)$

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