# 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

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