If the truth value of the statement p → (~ q ∨ r) is false (F), then the truth values of the statementsp, q, r are respectively

Correct answer: C. T, T, F Recall, p → q is false if and only if p is true and q is false Alternatively, Truth table of statement p → (~ q ∨ r) p q r ~ q ~ q ∨ r p → (~ q ∨ r) T T T F T T T T F F F F T F T T T T T F F T T T F T T F T T F T F F F T F F T T T T F F F T T T So, the truth values are p: true, q: true, r: false

JEE Main 201912 AprilMorning Shift

Exam: JEE Main
Year: 2019
Date: 12 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Mathematical Reasoning

JEE Main 2019 · 12 April · Morning Shift · Mathematics · Mathematical Reasoning

If the truth value of the statement p → (~ q ∨ r) is false (F),...

If the truth value of the statement

p \to \left(~ q ∨ r\right) is false \left(F\right), then

the truth values of the statementsp, q, r

are respectively

Options

A $T, F, T$
B $T, F, F$
C $T, T, F$
D $F, T, T$

Correct answer

C. $T, T, F$

Step-by-step solution

Recall, $p \to q$ is false if and only if $p$ is true and $q$ is false

Alternatively,

Truth table of statement $p \to \left(~ q ∨ r\right)$

$p$	$q$	$r$	$~ q$	$~ q ∨ r$	$p \to \left(~ q ∨ r\right)$
$T$	$T$	$T$	$F$	$T$	$T$
$T$	$T$	$F$	$F$	$F$	$F$
$T$	$F$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$T$	$T$
$F$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$T$	$T$
$F$	$F$	$F$	$T$	$T$	$T$

So, the truth values are

p:

true,

q:

true,

r:

false

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$p$	$q$	$r$	$~ q$	$~ q ∨ r$	$p \to \left(~ q ∨ r\right)$
$T$	$T$	$T$	$F$	$T$	$T$
$T$	$T$	$F$	$F$	$F$	$F$
$T$	$F$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$T$	$T$
$F$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$T$	$T$
$F$	$F$	$F$	$T$	$T$	$T$

$p$	$q$	$r$	$~ q$	$~ q ∨ r$	$p \to \left(~ q ∨ r\right)$
$T$	$T$	$T$	$F$	$T$	$T$
$T$	$T$	$F$	$F$	$F$	$F$
$T$	$F$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$T$	$T$
$F$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$T$	$T$
$F$	$F$	$F$	$T$	$T$	$T$

$p$	$q$	$r$	$~ q$	$~ q ∨ r$	$p \to \left(~ q ∨ r\right)$
$T$	$T$	$T$	$F$	$T$	$T$
$T$	$T$	$F$	$F$	$F$	$F$
$T$	$F$	$T$	$T$	$T$	$T$
$T$	$F$	$F$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$T$	$T$
$F$	$T$	$F$	$F$	$F$	$T$
$F$	$F$	$T$	$T$	$T$	$T$
$F$	$F$	$F$	$T$	$T$	$T$