Consider : Statement - I : (p ∧ ~ q) ∧ (~ p ∧ q) is a fallacy. Statement - II : (p → q) ↔ (~ q → ~ p) is a tautology.

Correct answer: D. Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for Statement - I . The truth table for Statement I is p q ~ p ~ q p ∧ ~ q ~ p ∧ q (p ∧ ~ q) ∧ (~ p ∧ q) TTFF TFTF FFTT FTFT FTFF FFTF FFFF Hence, it is a fallacy. The truth table for statement II is p q ~ p ~ q p → q ~ q → ~ p (p → q) ↔ (~ q → ~ p) TTFF TFTF FFTT FTFT TFTT TFTT TTTT Hence, it is a tautology. But, the second statement is not a correct explanation for the first.

JEE Main 20137 April

Exam: JEE Main
Year: 2013
Date: 7 April
Subject: Mathematics
Chapter: Mathematical Reasoning

JEE Main 2013 · 7 April · Mathematics · Mathematical Reasoning

Consider : Statement - I : (p ∧ ~ q) ∧ (~ p ∧ q) is a fallacy....

Consider :
Statement - $I$ : \left(p ∧ ~ q\right) ∧ \left(~ p ∧ q\right) is a fallacy.
Statement - $II$ : \left(p \to q\right) ↔ \left(~ q \to ~ p\right) is a tautology.

Options

AStatement - $I$ is true; statement - $II$ is false.
BStatement - $I$ is false; Statement - $II$ is true.
CStatement - $I$ true; Statement - $II$ is true; Statement - $II$ is a correct explanation for Statement - $I$ .
DStatement - $I$ is true; Statement - $II$ is true; Statement - $II$ is not a correct explanation for Statement - $I$ .

Correct answer

D. Statement - $I$ is true; Statement - $II$ is true; Statement - $II$ is not a correct explanation for Statement - $I$ .

Step-by-step solution

The truth table for Statement $I$ is

$p$	$q$	$~ p$	$~ q$	$p ∧ ~ q$	$~ p ∧ q$	\left(p ∧ ~ q\right) ∧ \left(~ p ∧ q\right)
$TTFF$	$TFTF$	$FFTT$	$FTFT$	$FTFF$	$FFTF$	$FFFF$

Hence, it is a fallacy.

The truth table for statement $II$ is

$p$	$q$	$~ p$	$~ q$	$p \to q$	$~ q \to ~ p$	\left(p \to q\right) ↔ \left(~ q \to ~ p\right)
$TTFF$	$TFTF$	$FFTT$	$FTFT$	$TFTT$	$TFTT$	$TTTT

Hence, it is a tautology.

But, the second statement is not a correct explanation for the first.

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