Question

The Boolean expression $\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)$ is equivalent to
A.$\sim p \vee q$
B.$\sim q \vee p$
C.$p \vee q$
D.$\left( { \sim p} \right) \vee \left( { \sim q} \right)$

Answer 1

Here we need to find the expression which is equivalent to the given Boolean expression. Here we will use the truth table to find an equivalent solution of the given Boolean expression. We will make two tables, in which the first table will include the given Boolean expression and the second table will contain the equivalent expression of the given Boolean expression.

Complete step-by-step answer:
The given Boolean expression is $\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)$.
Now, we will draw our first truth table of this Boolean expression.
First column will include the term $p$ , second column will contain $q$ , third column will contain $p \wedge q$, fourth column will contain $\sim q$ , fifth column will contain $\left( { \sim q} \right) \vee p$ and the last column will contain resultant Boolean expression i.e. $\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)$.

$p$	$q$	$p \wedge q$	$\sim q$	$\left( { \sim q} \right) \vee p$	$\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)$
T	T	T	F	T	T
T	F	F	T	T	T
F	T	F	F	F	F
F	F	F	T	T	T

Now, we will draw the second truth table in which the first column will include the term $p$ , second column will contain $q$ , third column will contain $\sim p$ and the fourth column will contain $\sim p \vee q$

$p$	$q$	$\sim p$	$\sim p \vee q$
T	T	F	T
T	F	F	F
F	T	T	T
F	F	T	T

We can see from both these table that the given Boolean expression $\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)$ is equivalent to $\sim p \vee q$.
Hence, the correct option is option A.

Note: Since we have used the Boolean algebra here. Boolean algebra is defined as the category of algebra in which the variable’s values are the truth values i.e. true and false and the operations which are used or performed in Boolean algebra are – conjunction, disjunction and negation.