Question

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

Answer 1

Here the Boolean expression is given that is $\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$. Boolean expression is a logical statement that returns either true or false. We need to draw the Boolean table.

Complete step-by-step answer:
Boolean expression is the expression of logic. It deals with variables that can have two discrete values, $0$ means False $\left( {\text{F}} \right)$ and $1$ means True $\left( {\text{T}} \right)$. So here, Boolean expression is given
$\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$.
So first find out $\sim \left( {p \vee q} \right)$.
Now, let us draw the table.
Here the symbol $\vee$ is of OR and we follow additional property. If $p$ is true, then it is treated as $1$ and if $q$ is false, it is treated as $0$. Then, $p \vee q = p + q = 1 + 0 = 1$.

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

$\sim$ is the negation mark. For example, if we get $p \vee q$ as true , then $\sim \left( {p \vee q} \right)$ must be false.
Now let us find $\left( { \sim p \wedge q} \right)$.
Here $\wedge$ is the symbol of AND and we follow multiplication property.

$p$	$\sim p$	$q$	$p \wedge q$	$\sim p \wedge q$
${\text{T}}$	${\text{F}}$	${\text{T}}$	${\text{T}}$	${\text{F}}$
${\text{T}}$	${\text{F}}$	${\text{F}}$	${\text{F}}$	${\text{F}}$
${\text{F}}$	${\text{T}}$	${\text{T}}$	${\text{F}}$	${\text{T}}$
${\text{F}}$	${\text{T}}$	${\text{F}}$	${\text{F}}$	${\text{F}}$

Now we have to find $\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$
Now let us arrange the table

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

Here we got just opposite of $p$. So the answer is $\sim p$.

So option C is the correct answer.

Note: If we write $p \wedge q$, then it is an AND operator but if $\sim \left( {p \wedge q} \right)$is given, then it becomes a NAND operator. Now, if we write $p \vee q$, then it is an OR operator and similarly, if $\sim \left( {p \vee q} \right)$is given, then it becomes a NOR operator.