The negation of the statement $(p \wedge q) \rightarrow(\sim p \vee r)$ is

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

Solution

$\begin{aligned} & \sim[(p \wedge q) \rightarrow(\sim p \vee r)] \\ & \equiv(p \wedge q) \wedge \sim(\sim p \vee r) \ldots[\because \sim(p \rightarrow q) \equiv p \wedge \sim q] \\ & \equiv p \wedge q \wedge p \wedge \sim r \quad \ldots[\text { Associative Law }] \\ & \equiv p \wedge q \wedge \sim r \quad \ldots[\text { Idempotent Law }]\end{aligned}$

Asked in: MHT CET 2023 (09 May Shift 1)

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