The negation of $\mathrm{p} \wedge(\mathrm{q} \rightarrow \mathrm{r})$ is

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

Solution

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

Asked in: MHT CET 2021 (23 Sep Shift 1)

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