For the statements p and q, consider the following compound statements: (a)(~ q ∧ (p → q)) → ~ p (b)((p ∨ q) ∧ ~ p) → q Then which of the following statements is correct?

Correct answer: B. (a) and (b) both are tautologies. p q ~ q p → q ~ p (~ q ∧ (p → q) ) T T F T F F T T F T F F F T F T F T T F T F F T T T T T p q p ∨ q ~ p (p ∨ q) ∧ ~ p T T T F F T T F T F F T F T T T T T F F F T F T Both are tautologies

JEE Main 202124 FebruaryEvening Shift

Exam: JEE Main
Year: 2021
Date: 24 February
Shift: Evening Shift
Subject: Mathematics
Chapter: Mathematical Reasoning

JEE Main 2021 · 24 February · Evening Shift · Mathematics · Mathematical Reasoning

For the statements p and q, consider the following compound...

For the statements $p and q,$ consider the following compound statements:

\left(a\right)\left(~ q ∧ \left(p \to q\right)\right) \to ~ p $</p><p>\left(b\right)\left(\left(p ∨ q\right) ∧ ~ p\right) \to q$

Then which of the following statements is correct?

Options

A\left(b\right) $is a tautology but not \left(a\right)$ .
B $\left(a\right)$ and $\left(b\right)$ both are tautologies.
C $\left(a\right)$ and $\left(b\right)$ both are not tautologies.
D\left(a\right) $is a tautology but not \left(b\right)$ .

Correct answer

B. $\left(a\right)$ and $\left(b\right)$ both are tautologies.

Step-by-step solution

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

$</p><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td>$ p $</td> <td>$ q $</td> <td>$ p ∨ q $</td> <td>$ ~ p $</td> <td>\left(p ∨ q\right)$ ∧ ~ p $</td> </tr> <tr> <td>$ T $</td> <td>$ T $</td> <td>$ T $</td> <td>$ F $</td> <td>$ F $</td> <td>$ T $</td> </tr> <tr> <td>$ T $</td> <td>$ F $</td> <td>$ T $</td> <td>$ F $</td> <td>$ F $</td> <td>$ T $</td> </tr> <tr> <td>$ F $</td> <td>$ T $</td> <td>$ T $</td> <td>$ T $</td> <td>$ T $</td> <td>$ T $</td> </tr> <tr> <td>$ F $</td> <td>$ F $</td> <td>$ F $</td> <td>$ T $</td> <td>$ F $</td> <td>$ T

Both are tautologies

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