Question

Which of the following statements is a tautology?

• A. $(~q\wedge p)\wedge q$
• B. $(~q\wedge p)\wedge (p\wedge ~p)$
• C. $(~q\wedge p)\vee (p\vee ~p)$
• D. $\left(p\wedge q\right)\wedge \left(~\left(p\wedge q\right)\right)$

Answer 1

The correct option is C

$(~q\wedge p)\vee (p\vee ~p)$

Explanation for correct option:

A tautology is a compound statement that is always true.

Signs of tautology:

Symbol	Meaning
$~$	Not
$\wedge$	And
$\vee$	Or
$\to$	Implies or If-then

Make truth table to check the relation $(~q\wedge p)\vee (p\vee ~p)$

$p$	$~p$	$q$	$~q$	$(~q\wedge p)$	$(p\vee ~p)$	$(~q\wedge p)\vee (p\vee ~p)$
$T$	$F$	$T$	$F$	$F$	$T$	$T$
$T$	$F$	$F$	$T$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$F$	$T$	$T$
$F$	$T$	$F$	$T$	$F$	$T$	$T$

Since the results are true. $(~q\wedge p)\vee (p\vee ~p)$ is tautology.

Explanation for incorrect options:

Option (A):

$p$	$~p$	$q$	$~q$	$(~q\wedge p)$	$(~q\wedge p)\wedge q$
$T$	$F$	$T$	$F$	$F$	$F$
$T$	$F$	$F$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$F$	$F$
$F$	$T$	$F$	$T$	$F$	$F$

Since all results are not true. So $(~q\wedge p)\wedge q$ is not a tautology.

Option (B):

$p$	$~p$	$q$	$~q$	$(~q\wedge p)$	$(p\vee ~p)$	$(~q\wedge p)\wedge (p\wedge ~p)$
$T$	$F$	$T$	$F$	$F$	$T$	$F$
$T$	$F$	$F$	$T$	$T$	$T$	$T$
$F$	$T$	$T$	$F$	$F$	$T$	$F$
$F$	$T$	$F$	$T$	$F$	$T$	$F$

Since all results are not true. So $(~q\wedge p)\wedge (p\wedge ~p)$ is not a tautology.

Option (D):

$p$	$~p$	$q$	$~q$	$\left(p\wedge q\right)$	$\left(~\left(p\wedge q\right)\right)$	$\left(p\wedge q\right)\wedge \left(~\left(p\wedge q\right)\right)$
$T$	$F$	$T$	$F$	$T$	$F$	$F$
$T$	$F$	$F$	$T$	$F$	$T$	$F$
$F$	$T$	$T$	$F$	$F$	$T$	$F$
$F$	$T$	$F$	$T$	$F$	$T$	$F$

Since all results are false. So, it is a contradiction.

Thus it is not a tautology

Hence option (C) is correct.