Question

Which of the following Boolean expression is a tautology?

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

Answer 1

The correct option is D

$\left(p\wedge q\right)\to \left(p\to q\right)$

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.

Explanation for the correct option(D):

The truth table is:

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

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

So, it is a tautology.

Hence, the correct option is $D)\left(p\wedge q\right)\to \left(p\to q\right)$

Explanation for the incorrect options:

The truth table is:

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

All outputs of options $A),B),C)$ are not true.

So, these options are incorrect.