Question

Which of the following statements is a tautology?

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

Answer 1

The correct option is C

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

Explanation for the correct option:

A tautology is a compound statement in Maths that always results in a Truth value.

It doesn’t matter what the individual part consists of, the result in tautology is always true.

The possible truth table is:

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

All outputs of $\left(~q\wedge p\right)\wedge \left(p\vee ~p\right)$ are true.

Hence, the correct option is $C)\left(~q\wedge p\right)\wedge \left(p\vee ~p\right)$

Explanation for the incorrect options:

The possible truth table is:

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

The outputs of the options $A),B),D)$ are not a tautology.

Hence, these options are incorrect.

Hence option (C) is the correct option.