Question

Which one of the following Boolean expressions is a tautology?
$\begin{aligned} & \text{A}\text{. }\left( p\vee q \right)\wedge \left( \sim p\vee \sim q \right) \\\ & \text{B}\text{. }\left( p\wedge q \right)\vee \left( p\vee \sim q \right) \\\ & \text{C}\text{. }\left( p\vee q \right)\wedge \left( p\vee \sim q \right) \\\ & \text{D}\text{. }\left( p\vee q \right)\vee \left( p\vee \sim q \right) \\\ \end{aligned}$

Answer 1

To solve this question we have to draw a truth table for all the expressions given in the options. The output values of all the possible combinations of the expression must be true for expression to be a tautology. So, the expression from the options gives the output values true is a tautology.

Complete step-by-step answer :
Before solving this question let us first understand the meaning of the symbols used in the question.
Here, symbol $\wedge$ represents the AND operator, symbol $\vee$ represents the OR operator and symbol $\sim$ represents the NOT operator.
Now, let us draw the truth table for the option A i.e. $\left( p\vee q \right)\wedge \left( \sim p\vee \sim q \right)$

$p$	$q$	$\sim p$	$\sim q$	$\left( p\vee q \right)$	$\left( \sim p\vee \sim q \right)$	$\left( p\vee q \right)\wedge \left( \sim p\vee \sim q \right)$
True	True	False	False	True	False	False
True	False	False	True	True	True	True
False	True	True	False	True	True	True
False	False	True	True	False	True	False

Now, we will draw a truth table for option B, i.e. $\left( p\wedge q \right)\vee \left( p\vee \sim q \right)$

$p$	$q$	$\sim p$	$\sim q$	$\left( p\wedge q \right)$	$\left( p\vee \sim q \right)$	$\left( p\wedge q \right)\vee \left( p\vee \sim q \right)$
True	True	False	False	True	True	True
True	False	False	True	False	True	True
False	True	True	False	False	False	False
False	False	True	True	False	True	True

Now, we will draw a truth table for option C, i.e. $\left( p\vee q \right)\wedge \left( p\vee \sim q \right)$

$p$	$q$	$\sim p$	$\sim q$	$\left( p\vee q \right)$	$\left( p\vee \sim q \right)$	$\left( p\vee q \right)\wedge \left( p\vee \sim q \right)$
True	True	False	False	True	True	True
True	False	False	True	True	True	True
False	True	True	False	True	False	False
False	False	True	True	False	True	False

Now, we will draw a truth table for option D, i.e. $\left( p\vee q \right)\vee \left( p\vee \sim q \right)$

$p$	$q$	$\sim p$	$\sim q$	$\left( p\vee q \right)$	$\left( p\vee \sim q \right)$	$\left( p\vee q \right)\vee \left( p\vee \sim q \right)$
True	True	False	False	True	True	True
True	False	False	True	True	True	True
False	True	True	False	True	False	True
False	False	True	True	False	True	True

It is clear from the truth tables that only expression $\left( p\vee q \right)\vee \left( p\vee \sim q \right)$ gives all True values in output, so $\left( p\vee q \right)\vee \left( p\vee \sim q \right)$ is a tautology.
Hence, option D is the correct answer.

Note : Be careful while solving AND operator and OR operator because students get confused between the two symbols and make mistakes. It is necessary to check all options because sometimes a question has multiple correct options.