If the Boolean expression
Question:

If the Boolean expression $(\mathrm{p} \Rightarrow \mathrm{q}) \Leftrightarrow(\mathrm{q} *(\sim \mathrm{p}))$ is a tautology, then the Boolean expression $\mathrm{p} *(\sim \mathrm{q})$ is equivalent to

1. (1) $\mathrm{q} \Rightarrow \mathrm{p}$

2. (2) $\sim \mathrm{q} \Rightarrow \mathrm{p}$

3. (3) $\mathrm{p} \Rightarrow \sim \mathrm{q}$

4. (4) $\mathrm{p} \Rightarrow \mathrm{q}$

Correct Option: 1,

Solution:

$\because p \rightarrow q \equiv \sim p \vee q$

$\mathrm{So}, * \equiv \mathrm{v}$

Thus, $\mathrm{p}^{*}(\sim \mathrm{q}) \equiv \mathrm{pv}(\sim \mathrm{q})$

$\equiv \mathrm{q} \rightarrow \mathrm{p}$