JEE Main 2023 — Mathematical Reasoning Question with Solution

Given condition is, $p\wedge q\to \left(r\vee q\right)\wedge \left(\left(p\wedge r\right)\to q)\right).....\left(1\right)$

We know conditional disjunction as,

$p\to q\equiv ~p\vee q$

Hence, $\left(p\wedge r\right)\to q\equiv ~\left(p\wedge r\right)\vee q$

Using De Morgan's law

$\left(p\wedge r\right)\to q\equiv ~\left(p\wedge r\right)\vee q\equiv \left(~p\vee ~r\right)\vee q$

Equation $\left(1\right)$ can be written as,

$\Rightarrow p\wedge q\to \left(r\vee q\right)\wedge \left(\left(~p\vee ~r\right)\vee q)\right)$

$\Rightarrow ~\left(p\wedge q\right)\vee \left(r\vee q\right)\wedge \left(\left(~p\vee ~r\right)\vee q)\right)$

$\Rightarrow \left(~p\vee ~q\right)\vee \left(r\vee q\right)\wedge \left(\left(~p\vee ~r\right)\vee q)\right)$

$\Rightarrow (~p\vee \left(~q\vee r\vee q\right)\wedge \left(\left(~p\vee ~r\right)\vee q)\right)$

$\Rightarrow \left(~p\vee t\right)\wedge \left(\left(~p\vee ~r\right)\vee q)\right)$

$\Rightarrow ~p\vee ~r\vee q$

For above statement to be a tautology,

$r$ can be $~p$ or $q$

When $r=~p$ or $r=q\Rightarrow ~p\vee ~r\vee q=t$

$\therefore$ Two value of $r$ are possible