JEE Main 2022 — Mathematical Reasoning Question with Solution

$$$~\left(p\iff ~q\right)\wedge q$  $=\left(p\iff q\right)\wedge q$

Proof: Solving the given $~\left(p\iff ~q\right)$

$=~\left(\left(p\to ~q\right)\wedge \left(~q\to p\right)\right)$

$=~\left(\left(~p\vee ~q\right)\wedge \left(q\vee p\right)\right)$

$=~\left(~\left(p\wedge q\right)\wedge \left(p\vee q\right)\right)$

$=\left(p\wedge q\right)\vee ~\left(p\vee q\right) ...\left(1\right)$

Now, $p\iff q\equiv \left(p\to q\right)\wedge \left(q\to p\right)$

$=\left(~p\vee q\right)\wedge \left(~q\vee p\right)$

$=\left(~p\vee q\right)\wedge ~q\vee \left(~p\vee q\right)\wedge p$

$=\left(~p\wedge ~q\right)\vee \left(q\wedge ~q\right)\vee \left(\left(~p\wedge p\right)\vee \left(q\wedge p\right)\right)$

$=\left(~p\wedge ~q\right)\vee c\vee \left(c\vee \left(q\wedge p\right)\right)$ $\left\{\text{where c is contradiction}\right\}$

$=~p\wedge ~q\vee \left(p\wedge q\right)$

$=~\left(p\vee q\right)\vee \left(p\wedge q\right) ...\left(2\right)$

So from equation $\left(1\right) \& \left(2\right)$ we get

$~\left(p\iff ~q\right)\wedge q\equiv \left(p\iff q\right)\wedge q$

$p$	$q$	$p\leftrightarrow q$	$\left(p\leftrightarrow q\right)\wedge q$	$\left(p\to q\right)$	$\left(p\to q\right)\wedge q$	$\left(p\to 1\right)\wedge p$
$T$	$T$	$T$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$F$	$F$	$F$	$F$
$F$	$T$	$F$	$F$	$T$	$T$	$F$
$F$	$F$	$T$	$F$	$T$	$F$	$F$

$\therefore \left(~\left(p\iff ~q\right)\right)\wedge q$ is equivalent to $\left(p\Rightarrow q\right)\wedge p$.