Gate CS 2021 (Set 1)| Question - 7 | Discrete Mathematics

$S_1: (\neg P\wedge (p\vee q\left)\right)\to q$
$S_2: q\to (\neg p\wedge (p\vee q\left)\right)$

Solve S₁:
_{$S_1: (\neg p\wedge (p\vee q\left)\right)\to q$}
$\equiv \left[\overline{p}\right(p+q\left)\right]\to q$
$\equiv (\overline{p}p+\overline{p}q)\to q$
$\equiv \overline{p}q\to q$
$\equiv \overline{\left(\overline{p}q\right)}+q$
$\equiv p+\overline{q}+q$
$\equiv p+1$ (Tautology)

Solve S₂:
$S_2: q\to (\neg p\wedge (p\vee q\left)\right)$
$\equiv q\to \left[\overline{p}\right(p+q\left)\right]$
$\equiv q\to (\overline{p}p+\overline{p}q)$
$\equiv q\to \overline{p}q$
$\equiv \overline{q} + \overline{p}q$ (Contingency, Not a Tautology)

So the correct answer is B.