JEE Main 202324 JanuaryMorning Shift

Exam: JEE Main
Year: 2023
Date: 24 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Mathematical Reasoning

JEE Main 2023 · 24 January · Morning Shift · Mathematics · Mathematical Reasoning

The compound statement ( ~ ( P ∧ Q ) ) ∨ ( ( ~ P ) ∧ Q ) ⇒ ( ( ~ P...

The compound statement

( ~ ( P ∧ Q ) ) ∨ ( ( ~ P ) ∧ Q ) ⇒ ( ( ~ P ) ∧ ( ~ Q ) )

is equivalent to

Options

A
$( ( ~ P ) ∨ Q ) ∧ ( ( ~ Q ) ∨ P )$
B $( ~ Q ) ∨ P$
C $( ( ~ P ) ∨ Q ) ∧ ( ~ Q )$
D $( ~ P ) ∨ Q$

Correct answer

$( ( ~ P ) ∨ Q ) ∧ ( ( ~ Q ) ∨ P )$

Step-by-step solution

We have,

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

Also,

P

Q

~ P

~ Q

\left(~ P ∨ Q\right)

</td> <td style="text-align: center;">\left(\left(~ Q\right) ∨ P\right)

\left(\left(~ P\right) ∧ Q\right) ∧ \left(\left(~ Q\right) ∨ P\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;">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;">T</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;">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;">F</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;">T</td> <td style="text-align: center;">T</td> <td style="text-align: center;">T</td> </tr> </tbody></table><p>Hence,

\left(~ \left(P ∧ Q\right)\right) ∨ \left(\left(~ P\right) ∧ Q\right) ⇒ \left(\left(~ P\right) ∧ \left(~ Q\right)\right) ≡ \left(\left(~ P\right) ∧ Q\right) ∧ \left(\left(~ Q\right) ∨ P\right).

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