Question

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

Answer 1

Hint: The given Boolean expression is \[ \sim \left( {p \Rightarrow \left( { \sim q} \right)} \right)\] when we have to interpret this Boolean expression firstly interpret the meaning of implies ($ \Rightarrow $) sign then after interpreting negation $ \sim $ sign we get the required expression which is equivalent to the given Boolean expression.

Complete step-by-step answer:
Here, The given Boolean expression is \[ \sim \left( {p \Rightarrow \left( { \sim q} \right)} \right)\] .
$p \Rightarrow \left( { \sim q} \right)$ is interpreted as $\left( { \sim p \vee \sim q} \right)$
Now, the Boolean expression \[ \sim \left( {p \Rightarrow \left( { \sim q} \right)} \right)\] can be written as $ \sim \left( { \sim p \vee \sim q} \right)$
$ \sim \left( { \sim p \vee \sim q} \right)$ can be interpreted as $ \sim \left( { \sim p} \right) \wedge \sim \left( { \sim q} \right)$
By applying double complement law we can write
$ \sim \left( { \sim p} \right) \wedge \sim \left( { \sim q} \right)$ is equivalent to $p \wedge q$
Thus, the given Boolean expression \[ \sim \left( {p \Rightarrow \left( { \sim q} \right)} \right)\] is equivalent to $p \wedge q$ .

Hence, option (D) is correct for this question.

Note: Double complement law:- This law states that negation of the negation of any quantity (double negation) gives the actual quantity. This can be mathematically represented as $ \sim \left( { \sim x} \right) = x$ .
While solving the problem related to the Boolean expression it is suggested to interpret first the inner Boolean operator then the other operators.
Now some important Boolean operators are
1. ‘And’ operator:- This operator tells us that a particular quantity is present in the two given sets. Mathematically this can be written as $\left( {x \wedge y} \right)$ which means a quantity must be present in set $x$ and set $y$.
2. ‘Or’ operator:- This operator tells us that a quantity is either present in one set or other sets. Mathematically this can be written as $\left( {x \vee y} \right)$ which is interpreted as a quantity that is either present in set $x$ or set $y$.
3. ‘Not’ operator:- This operator tells that the quantity is not present in the set. Mathematically this can be written as $\left( { \sim x} \right)$ which means a quantity is not present in the set $x$.