The proposition $(\sim p) \vee(p \wedge \sim q)$ is equivalent to

The proposition $(\sim p) \vee(p \wedge \sim q)$ is equivalent to
  1. $\mathrm{p} \wedge(\sim \mathrm{q})$
  2. $\mathrm{p} \rightarrow(\sim \mathrm{q})$
  3. $\mathrm{p} \vee(\mathrm{q})$
  4. $\mathrm{q} \rightarrow \mathrm{p}$

Solution

$\begin{array}{ll}(\sim p) \vee(p \wedge \sim q) & \\ \equiv(\sim p \vee p) \wedge(\sim p \vee \sim q) & \ldots[\text { Distributive law }] \\ \equiv T \wedge(\sim p \vee \sim q) & \ldots[\text { Complement law }] \\ \equiv \sim p \vee \sim q & \ldots[\text { Identity law] } \\ \equiv p \rightarrow(\sim q) & \ldots[\because p \rightarrow q \equiv \sim p \vee q]\end{array}$

Asked in: MHT CET 2024 (16 May Shift 1)

Practice more Mathematical Logic questions on Aicharya