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

Given $\left(p ∧ r\right) ⇔ \left(p ∧ \left(~ q\right)\right) ≡ \left(~ p\right)$

Assuming all option one by one,

Taking $r = p$

$p$	$q$	$~ p$	$~ q$	$p ∧ p$	$p ∧ ~ q$	\left(p ∧ r\right) ⇔ \left(p ∧ \left(~ q\right)\right)$</td> </tr> <tr> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> </tr> <tr> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> </tr> <tr> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> </tr> <tr> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> </tr> </tbody></table><p>Now taking$r = ~ p$</p><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td style="text-align: center;">$p$</td> <td style="text-align: center;">$q$</td> <td style="text-align: center;">$~ p$</td> <td style="text-align: center;">$~ q$</td> <td style="text-align: center;">$p ∧ ~ p$</td> <td style="text-align: center;">$p ∧ ~ q$</td> <td style="text-align: center;">\left(p ∧ r\right) ⇔ \left(p ∧ \left(~ q\right)\right)$
$T$	$T$	$F$	$F$	$F$	$F$	$T$
$T$	$F$	$F$	$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

Taking $r = q$

$p$

$q$

$~ p$

$~ q$

$p ∧ q$

$p ∧ ~ q$

\left(p ∧ r\right) ⇔ \left(p ∧ \left(~ q\right)\right)$</td> </tr> <tr> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> </tr> <tr> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> </tr> <tr> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> </tr> <tr> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$T$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$F$</td> <td style="text-align: center;">$T$</td> </tr> </tbody></table><p>So clear, \left(p ∧ r\right) ⇔ \left(p ∧ \left(~ q\right)\right) ≡ \left(~ p\right)$ Practice Mathematical Reasoning on Quantrex Academy → More from Mathematical Reasoning Negation of p ∧ (q ∧ ~ (p ∧ q)) is JEE Main 2023The statement ( p ∧ ( ~ q ) ) ∨ ( ( ~ p ) ∧ q ) ∨ ( ( ~ p ) ∧ ( ~ q ) ) is equivalent to… JEE Main 2023The negation of the statement ( ( A ∧ ( B ∨ C ) ) ⇒ ( A ∨ B ) ) ⇒ A is JEE Main 2023Among the two statements ( S _ 1 ): (p ⇒ q) ∧ (p ∧ (~ q)) is a contradiction and ( S _ 2… JEE Main 2023The converse of ((~ p) ∧ q) ⇒ r is JEE Main 2023The number of ordered triplets of the truth values of p, q and r such that the truth… JEE Main 2023The statement ~ [p ∨ (~ (p ∧ q))] is equivalent to JEE Main 2023The negation of the statement (p ∨ q) ∧ (q ∨ (~ r)) is JEE Main 2023The negation of ( p ∧ ( - q ) ) ∨ ( - p ) is equivalent to JEE Main 2023Negation of (p → q) → (q → p) is JEE Main 2023Among the statements (S1): (p ⇒ q) ∨ ((~ p) ∧ q) is a tautology (S2): (q ⇒ p) ⇒ ((~ p) ∧… JEE Main 2023Statement ( P ⇒ Q ) ∧ ( R ⇒ Q ) is logically equivalent to JEE Main 2023Which of the following statements is a tautology? JEE Main 2023The negation of the expression q ∨ ( ( ~ q ) ∧ p ) is equivalent to JEE Main 2023The number of values of r ∈ \ p, q, ~ p, ~ q\ for which ((p ∧ q) ⇒ (r ∨ q) ∧ ( (p ∧ r) ⇒… JEE Main 2023Consider: (S1): (p ⇒ q) ∨ (p ∧ (~ q)) is a tautology. (S2): ((~ p) ⇒ (~ q)) ∧ ((~ p) ∨ q)… JEE Main 2023 Full Mathematical Reasoning list Mathematical Reasoning JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs