Let r ∈ (P, q, ~ p, ~ q) be such that the logical statement r ∨ (~ p) ⇒ (p ∧ q) ∨ r is a tautology. Then r is equal to

Correct answer: C. ~ p Solving all option one by one we get following truth table, (1) p = r q ~ p r ∨ (~ p) (p ∧ q) (p ∧ q) ∨ r r ∨ (~ p) ⇒ (p ∧ q) ∨ r T F F T F T T F T T T F F F T T F T T T T F F T T F F T (2) p ~ p r ∨ (~ p) q = r (p ∧ q) (p ∧ q) ∨ r r ∨ (~ p) ⇒ (p ∧ q) ∨ r T F T T T T T F T T T F T T T F F F F F T F T T F F F F (3) p q r = ~ p r ∨ (~ p) (p ∧ q) (p ∧ q) ∨ r r ∨ (~ p) ⇒ (p ∧ q) ∨ r T T F F T T T F T T T F T T T F F F F F T F F T T F T T (4) ~ p p q r ∨ (~ p) r = ~ q (p ∧ q) (p ∧ q) ∨ r r ∨ (~ p) ⇒ (p ∧ q) ∨ r F T T F F T T T F T F T T F T T T F T T F F F F T F F T T F T T Now final answer is option C.

JEE Main 202226 JuneEvening Shift

Exam: JEE Main
Year: 2022
Date: 26 June
Shift: Evening Shift
Subject: Mathematics
Chapter: Mathematical Reasoning

JEE Main 2022 · 26 June · Evening Shift · Mathematics · Mathematical Reasoning

Let r ∈ (P, q, ~ p, ~ q) be such that the logical statement r ∨ (~...

Let

r \in

\left(P, q, ~ p, ~ q\right)

be such that the logical statement

r ∨ \left(~ p\right) ⇒ \left(p ∧ q\right) ∨ r

is a tautology. Then

r is equal to

Options

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

Correct answer

C. $~ p$

Step-by-step solution

Solving all option one by one we get following truth table,

(1)

$p = r$	$q$	$~ p$	$r ∨ \left(~ p\right)</td> <td style="text-align: center;">\left(p ∧ q\right)</td> <td style="text-align: center;">\left(p ∧ q\right) ∨ r$	$r ∨ \left(~ p\right) ⇒ \left(p ∧ q\right) ∨ r$
$T$	$F$	$F$	$T$	$F$	$T$	$T$
$F$	$T$	$T$	$T$	$F$	$F$	$F$
$T$	$T$	$F$	$T$	$T$	$T$	$T$
$F$	$F$	$T$	$T$	$F$	$F$	$T$

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

(3) $</p><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td style="text-align: center;">$ p $</td> <td style="text-align: center;">$ q $</td> <td style="text-align: center;">$ r = ~ p $</td> <td style="text-align: center;">$ r ∨ \left(~ p\right) \left(p ∧ q\right) \left(p ∧ q\right) ∨ r $</td> <td style="text-align: center;">$ r ∨ \left(~ p\right) ⇒ \left(p ∧ q\right) ∨ r $</td> </tr> <tr> <td style="text-align: center;">$ 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 $</td> <td style="text-align: center;">$ 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 $</td> <td style="text-align: center;">$ T $</td> <td style="text-align: center;">$ F $</td> <td style="text-align: center;">$ T $</td> <td style="text-align: center;">$ T $</td> </tr> <tr> <td style="text-align: center;">$ T $</td> <td style="text-align: center;">$ F $</td> <td style="text-align: center;">$ F $</td> <td style="text-align: center;">$ 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 $</td> <td style="text-align: center;">$ T $</td> <td style="text-align: center;">$ T $</td> </tr> </tbody></table><p>(4)$

$~ p$	$p$	$q$	$r ∨ \left(~ p\right)$	$r = ~ q$	\left(p ∧ q\right) $</td> <td style="text-align: center;">\left(p ∧ q\right) ∨ r$	$r ∨ \left(~ p\right) ⇒ \left(p ∧ q\right) ∨ r$
$F$	$T$	$T$	$F$	$F$	$T$	$T$	$T$
$F$	$T$	$F$	$T$	$T$	$F$	$T$	$T$
$T$	$F$	$T$	$T$	$F$	$F$	$F$	$F$
$T$	$F$	$F$	$T$	$T$	$F$	$T$	$T$

Now final answer is option C.

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