The negation of the statement pattern $\sim \mathrm{p} \vee(\mathrm{q} \rightarrow \sim \mathrm{r})$ is

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

Solution

$\begin{array}{l} \sim[\sim p \vee(q \rightarrow \sim r)] \\ \equiv p \wedge \sim(q \rightarrow \sim r) \\ \equiv p \wedge \sim(\sim q \vee \sim r) \\ \equiv p \wedge \sim[\sim(q \wedge r)] \\ \equiv p \wedge(q \wedge r) \end{array}$

Asked in: MHT CET 2020 (19 Oct Shift 2)

Practice more Mathematical Logic questions on Aicharya