p∧r⇔p∧~q is equivalent to ~p when r is

Correct answer: (C) q. Given p∧r⇔p∧~q≡~p Assuming all option one by one, Taking r=p p q ~p ~q p∧p p∧~q p∧r⇔p∧~q T T F F T F F T F F T T T T F T T F F F T F F T T F F T Now taking r=~p p q ~p ~q p∧~p p∧~q p∧r⇔p∧~q T T F F F F T T F F T F T F F T T F F F T F F T T F F T Taking r=q p q ~p ~q p∧q p∧~q p∧r⇔p∧~q T T F F T F F T F F T F T F F T T F F F T F F T T F F T So clear, p∧r⇔p∧~q≡~p

Q46 / 90 · JEE Main 2022 (27 Jul Shift 1) · Mathematics

$(p \land r) \Leftrightarrow (p \land (~ q))$ is equivalent to $(~ p)$ when $r$ is

A
$p$
B
$~ p$
Correct answer
C
$q$
D
$~ q$

Solution

Given $(p \land r) \Leftrightarrow (p \land (~ q)) \equiv (~ p)$
Assuming all option one by one,
Taking $r = p$

$p$ $q$ $~ p$ $~ q$ $p \land p$ $p \land ~ q$ $(p \land r) \Leftrightarrow (p \land (~ q))$

$T$ $T$ $F$ $F$ $T$ $F$ $F$

$T$ $F$ $F$ $T$ $T$ $T$ $T$

$F$ $T$ $T$ $F$ $F$ $F$ $T$

$F$ $F$ $T$ $T$ $F$ $F$ $T$

Now taking $r = ~ p$

$p$ $q$ $~ p$ $~ q$ $p \land ~ p$ $p \land ~ q$ $(p \land r) \Leftrightarrow (p \land (~ q))$

$T$ $T$ $F$ $F$ $F$ $F$ $T$

$T$ $F$ $F$ $T$ $F$ $T$ $F$

$F$ $T$ $T$ $F$ $F$ $F$ $T$

$F$ $F$ $T$ $T$ $F$ $F$ $T$

Taking $r = q$

$p$ $q$ $~ p$ $~ q$ $p \land q$ $p \land ~ q$ $(p \land r) \Leftrightarrow (p \land (~ q))$

$T$ $T$ $F$ $F$ $T$ $F$ $F$

$T$ $F$ $F$ $T$ $F$ $T$ $F$

$F$ $T$ $T$ $F$ $F$ $F$ $T$

$F$ $F$ $T$ $T$ $F$ $F$ $T$

So clear, $(p \land r) \Leftrightarrow (p \land (~ q)) \equiv (~ p)$

Hence, the correct answer is option: 'C'.

$p$	$q$	$~ p$	$~ q$	$p \land p$	$p \land ~ q$	$(p \land r) \Leftrightarrow (p \land (~ q))$
$T$	$T$	$F$	$F$	$T$	$F$	$F$
$T$	$F$	$F$	$T$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$~ p$	$~ q$	$p \land ~ p$	$p \land ~ q$	$(p \land r) \Leftrightarrow (p \land (~ q))$
$T$	$T$	$F$	$F$	$F$	$F$	$T$
$T$	$F$	$F$	$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

$p$	$q$	$~ p$	$~ q$	$p \land q$	$p \land ~ q$	$(p \land r) \Leftrightarrow (p \land (~ q))$
$T$	$T$	$F$	$F$	$T$	$F$	$F$
$T$	$F$	$F$	$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

p∧r⇔p∧~q is equivalent to ~p when r is

Solution

Relevant Questions

$(p \land r) \Leftrightarrow (p \land (~ q))$ is equivalent to $(~ p)$ when $r$ is