JEE Main 2022 — Mathematical Reasoning Question with Solution

Plotting the truth table of the following option we have,

$p$	$q$	$~p$	$~q$	$~p\vee q$
$T$	$T$	$F$	$F$	$T$
$T$	$F$	$F$	$T$	$F$
$F$	$T$	$T$	$F$	$T$
$F$	$F$	$T$	$T$	$T$

We know that, $A\Rightarrow B$ is only false when $A$ is true and $B$ is false, now using this rule and above table we will make truth table of options,

$\left(\left(~p\right)\vee q\right)\Rightarrow p$	$p\Rightarrow \left(\left(~p\right)\vee q\right)$	$\left(\left(~p\right)\vee q\right)\Rightarrow q$	$q\Rightarrow \left(\left(~p\right)\vee q\right)$
$T$	$T$	$T$	$T$
$T$	$F$	$T$	$T$
$F$	$T$	$T$	$T$
$F$	$T$	$F$	$T$

 

Hence $q\Rightarrow \left(\left(~p\right)\vee q\right)$ is tautology.