Consider the following two propositions : P _ 1 : ~ (p → ~ q) P _ 2 : (p ∧ ~ q) ∧ ((~ p) ∨ q) If the proposition p → ((~ p) ∨ q) is evaluated as FALSE, then

Correct answer: C. Both P _ 1 and P _ 2 are FALSE p q ~ p ~ q ~ p ∨ q p → (~ p ∨ q) p → ~ q ~ (p → ~ q) p ∧ ~ q (p ∧ ~ q) ∧ ((~ p) ∨ q) T T F F T T F T F F T F F T F F T F T F F T T F T T T F F F F F T T T T T F F F p → (~ p ∨ q) is F when p is true q is false from the table. Hence, P _ 1 and P _ 2 both are false

JEE Main 202225 JuneMorning Shift

Exam: JEE Main
Year: 2022
Date: 25 June
Shift: Morning Shift
Subject: Mathematics
Chapter: Mathematical Reasoning

JEE Main 2022 · 25 June · Morning Shift · Mathematics · Mathematical Reasoning

Consider the following two propositions : P _ 1 : ~ (p → ~ q) P _ 2...

Consider the following two propositions :
${P}_{1}$ : $~ \left(p \to ~ q\right)$
${P}_{2}$ : \left(p ∧ ~ q\right) ∧ \left(\left(~ p\right) ∨ q\right) $</p><p>If the proposition$ p \to \left(\left(~ p\right) ∨ q\right) is evaluated as FALSE, then

Options

A ${P}_{1}$ is TRUE and ${P}_{2}$ is FALSE
B ${P}_{1}$ is FALSE and ${P}_{2}$ is TRUE
CBoth ${P}_{1}$ and ${P}_{2}$ are FALSE
DBoth ${P}_{1}$ and ${P}_{2}$ are TRUE

Correct answer

C. Both ${P}_{1}$ and ${P}_{2}$ are FALSE

Step-by-step solution

p

q

~ p

~ q

~ p ∨ q

p \to \left(~ p ∨ q\right)

p \to ~ q

~ \left(p \to ~ q\right)

p ∧ ~ q

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

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p \to \left(~ p ∨ q\right)

is

when

is true

is false from the table.</p><p>Hence,

{P}_{1} \text{ and } {P}_{2} both are false

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