JEE Main 202329 JanuaryEvening Shift

Exam: JEE Main
Year: 2023
Date: 29 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Mathematical Reasoning

JEE Main 2023 · 29 January · Evening Shift · Mathematics · Mathematical Reasoning

The statement B ⇒ ( (~ A) ∨ B) is not equivalent to :

The statement

B ⇒

\left(

\left(~ A\right) ∨ B\right)

is not equivalent to :

Options

A $B ⇒ \left(A ⇒ B\right)$
B $A ⇒ \left(A ⇔ B\right)$
C $A ⇒ \left(\left(~ A\right) ⇒ B\right)$
D $B ⇒ \left(\left(~ A\right) ⇒ B\right)$

Correct answer

B. $A ⇒ \left(A ⇔ B\right)$

Step-by-step solution

Given statement is $B ⇒ \left(\left(~ A\right) ∨ B\right)$

Truth table for given statement:

$A$	$B$	$~ A$	$~ A ∨ B$	$B ⇒ \left(\left(~ A\right) ∨ B\right)$
$T$	$T$	$F$	$T$	$T$
$T$	$F$	$F$	$F$	$T$
$F$	$T$	$T$	$T$	$T$
$F$	$F$	$T$	$T$	$T$

It is a tautology.

Truth table of given options:

$A ⇒ B$	$~ A ⇒ B$	$B ⇒ \left(A ⇒ B\right)$	$A ⇒ \left(\left(~ A\right) ⇒ B\right)$	$B ⇒ \left(\left(~ A\right) ⇒ B\right)$
$T$	$T$	$T$	$T$	$T$
$F$	$T$	$T$	$T$	$T$
$T$	$T$	$T$	$T$	$T$
$T$	$F$	$T$	$T$	$T$

Option A: $B ⇒ \left(A ⇒ B\right)$ is a tautology.

Option C: $A ⇒ \left(\left(~ A\right) ⇒ B\right)$ is a tautology.

Option D: $B ⇒ \left(\left(~ A\right) ⇒ B\right) is a tautology.

Hence, it is very clear that given statement is not equivalent to option B.

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