Negation of the statement

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Question:

Negation of the statement $(p \vee r) \Rightarrow(q \vee r)$ is:

 

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

  2. $\sim \mathrm{p} \wedge \mathrm{q} \wedge \sim \mathrm{r}$

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

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

Correct Option: 1

Solution:

$\because \sim(\mathrm{A} \Rightarrow \mathrm{B})=\mathrm{A} \wedge \sim \mathrm{B}$

$\therefore \sim((p \vee r) \Rightarrow(q \vee r))$

$=(p \vee r) \wedge(\sim q \wedge \sim r)$

$=((p \vee r) \wedge(\sim r)) \wedge(\sim q)$

$=p \wedge(\sim r) \wedge(\sim q)$

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