For any two statements

Question:

For any two statements $\mathrm{p}$ and $\mathrm{q}$, the negation of the expression $p \vee(\sim p \wedge q)$ is

 

  1. $\mathrm{p} \wedge \mathrm{q}$

  2. $\mathrm{p} \leftrightarrow \mathrm{q}$

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

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


Correct Option: , 4

Solution:

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

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

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

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

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

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

 

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