# Which of the following is a tautology ?

Question:

Which of the following is a tautology ?

1. $(\sim \mathrm{p}) \wedge(\mathrm{p} \vee \mathrm{q}) \rightarrow \mathrm{q}$

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

3. $(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{p})$

4. $(\sim \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{q}$

Correct Option: 1

Solution:

Option (1) is

$\sim \mathrm{p} \wedge(\mathrm{p} \vee \mathrm{q}) \rightarrow \mathrm{q}$

$\equiv(\sim \mathrm{p} \wedge \mathrm{p}) \vee(\sim \mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{q}$

$\equiv \mathrm{C} \vee(\sim \mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{q}$

$\equiv(\sim \mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{q}$

$\equiv \sim(\sim \mathrm{p} \wedge \mathrm{q}) \vee \mathrm{q}$

$\equiv(p \vee \sim q) \vee q$

$\equiv(p \vee q) \vee(\sim q \vee q)$

$\equiv(p \vee q) \vee t$

so $\sim \mathrm{p} \wedge(\mathrm{p} \vee \mathrm{q}) \rightarrow \mathrm{q}$ is a tautology