Question

$ \sim \left[ {\left( { \sim p} \right) \wedge q} \right]$ is logically equivalent to
$\left( a \right) \sim \left( {p \vee q} \right)$
$\left( b \right) \sim \left[ {p \wedge \left( { \sim q} \right)} \right]$
$\left( c \right)p \wedge \left( { \sim q} \right)$
$\left( d \right)p \vee \left( { \sim q} \right)$
$\left( e \right)\left( { \sim p} \right) \vee \left( { \sim q} \right)$

Answer 1

Hint: In this particular question use the concept of distribution property i.e. $ \sim \left( {a \wedge b} \right) = \left( { \sim a} \right) \vee \left( { \sim b} \right)$ and according to this property simplify the given expression so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given expression
$ \sim \left[ {\left( { \sim p} \right) \wedge q} \right]$
Now according to the distributive property $ \sim \left( {a \wedge b} \right) = \left( { \sim a} \right) \vee \left( { \sim b} \right)$, so use this property in the above equation we have,
$ \Rightarrow \sim \left[ {\left( { \sim p} \right) \wedge q} \right] = \left[ { \sim \left( { \sim p} \right)} \right] \vee \left( { \sim q} \right)$................ (1)
Now as we know that $\left( \sim \right)$ it is the symbol of negation, that is the opposite of something.
So, $\left[ { \sim \left( { \sim p} \right)} \right]$ = opposite of, opposite of something which is equivalent to something i.e. p.
Therefore, $\left[ { \sim \left( { \sim p} \right)} \right]$ = p, so use this property in equation (1) we have,
$ \Rightarrow \sim \left[ {\left( { \sim p} \right) \wedge q} \right] = \left[ { \sim \left( { \sim p} \right)} \right] \vee \left( { \sim q} \right) = p \vee \left( { \sim q} \right)$
Therefore, $ \sim \left[ {\left( { \sim p} \right) \wedge q} \right] = p \vee \left( { \sim q} \right)$.
So, $ \sim \left[ {\left( { \sim p} \right) \wedge q} \right]$ is logically equivalent to $p \vee \left( { \sim q} \right)$.
So this is the required answer.

So, the correct answer is “Option d”.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the distribution property which is stated above and always recall that, $\left( \sim \right)$is the symbol of negation, that is opposite of something.