The negation of the Boolean expression

Question:

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

  1. $(\sim \mathrm{x} \wedge \mathrm{y}) \vee(\sim \mathrm{x} \wedge \sim \mathrm{y})$

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

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

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


Correct Option: , 3

Solution:

$p \leftrightarrow q \equiv(p \rightarrow q) \wedge(q \rightarrow p)$

$x \leftrightarrow \sim y \equiv(x \rightarrow \sim y) \wedge(-y \rightarrow x)$

$\because(p \rightarrow q \equiv \sim p \vee q)$

$\mathrm{X} \leftrightarrow \sim \mathrm{y}=(\sim \mathrm{X} \vee \sim \mathrm{y}) \wedge(\mathrm{y} \vee \mathrm{x})$

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

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