The negation of the Boolean expression

Question:

The negation of the Boolean expression $p \vee(\sim p \wedge q)$ is equivalent to :

  1. $\sim \mathrm{p} \vee \sim \mathrm{q}$

  2. $\sim \mathrm{p} \vee \mathrm{q}$

  3. $\sim \mathrm{p} \wedge \sim \mathrm{q}$

  4. $\mathrm{p} \wedge \sim \mathrm{q}$


Correct Option: , 3

Solution:

Negation of $\phi \vee(\sim p \wedge q)$

$\mathrm{p} \vee(\sim \mathrm{p} \wedge \mathrm{q})=(\mathrm{p} \vee \sim \mathrm{p}) \wedge(\mathrm{p} \vee \mathrm{q})$

$=(\mathrm{T}) \wedge(\mathrm{p} \vee \mathrm{q})$

$=(\mathrm{p} \vee \mathrm{q})$

now negation of $(\mathrm{p} \vee \mathrm{q})$ is

$\sim(\mathrm{p} \vee \mathrm{q})=\sim \mathrm{p} \wedge \sim \mathrm{q}$

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