Negation of $(p \wedge q) \rightarrow(\sim p \vee r)$ is

Negation of $(p \wedge q) \rightarrow(\sim p \vee r)$ is
  1. $\mathrm{p} \vee \mathrm{q} \vee(\sim \mathrm{r})$
  2. $\mathrm{p} \wedge \mathrm{q} \wedge \mathrm{r}$
  3. $\sim \mathrm{p} \wedge \mathrm{q} \wedge \mathrm{r}$
  4. $\mathrm{p} \wedge \mathrm{q} \wedge(\sim \mathrm{r})$

Solution

$\begin{aligned} & \sim[(\mathrm{p} \wedge \mathrm{q}) \rightarrow(\sim \mathrm{p} \vee \mathrm{r})] \\ & \equiv \sim[\sim(\mathrm{p} \wedge \mathrm{q}) \vee(\sim \mathrm{p} \vee \mathrm{r})] \\ & \equiv(\mathrm{p} \wedge \mathrm{q}) \wedge \sim(\sim \mathrm{p} \vee \mathrm{r}) \\ & \equiv(\mathrm{p} \wedge \mathrm{q}) \wedge(\mathrm{p} \wedge \sim \mathrm{r}) \equiv \mathrm{p} \wedge \mathrm{q} \wedge(\sim \mathrm{r}) \end{aligned}$

Asked in: MHT CET 2021 (20 Sep Shift 2)

Practice more Mathematical Logic questions on Aicharya