Question

The logical statement $\left[ \sim \left( \sim p\vee q \right)\vee \left( p\wedge r \right)\wedge \left( \sim q\wedge r \right) \right]$ is equivalent to:
(a) $\left( p\wedge r \right)\wedge \sim q$
(b) $\left( \sim p\wedge \sim q \right)\wedge r$
(c) $\sim p\vee r$
(d) $\left( p\wedge \sim q \right)\vee r$

Answer 1

Hint: First, before proceeding for this, we must know the following meanings of the operators which are logical in their sense. Then, by using the above definitions of the operators, we can solve the given statement by using the fact that $\sim \left( \sim a \right)=a$. Then, by using the property that $\left( a\wedge \sim b \right)\vee \left( a\wedge c \right)=a\wedge \left( \sim b\vee c \right)$ and then using the associative property of the logical operators as $a\wedge \left( \sim b\wedge c \right)=\left( a\wedge c \right)\wedge \sim b$and then by using it, we get the final result.

Complete step by step answer:
In this question, we are supposed to find the logical statement $\left[ \sim \left( \sim p\vee q \right)\vee \left( p\wedge r \right)\wedge \left( \sim q\wedge r \right) \right]$ is equivalent to which option.
So, before proceeding for this, we must know the following meanings of the operators which are logical in their sense as:
$\wedge $ is AND operator which means that only intersection of the given variables is the result.
$\vee $ is an OR operator which means that intersection and the remaining part of the given variables is the result.
$\sim $ is the NOT operator which means the negation of the given statement.
Now, by using the above definitions of the operators, we can solve the given statement by using the fact that $\sim \left( \sim a \right)=a$ as:
$\begin{align}
  & \left[ \sim \left( \sim p\vee q \right)\vee \left( p\wedge r \right)\wedge \left( \sim q\wedge r \right) \right] \\
 & \Rightarrow \left[ \left( p\wedge \sim q \right)\vee \left( p\wedge r \right)\wedge \left( \sim q\wedge r \right) \right] \\
\end{align}$
Then, by using the property that $\left( a\wedge \sim b \right)\vee \left( a\wedge c \right)=a\wedge \left( \sim b\vee c \right)$, we get:
$\left[ p\wedge \left( \sim q\vee r \right) \right]\wedge \left( \sim q\wedge r \right)$
Then, we know that the above statement is equivalent to the statement as:
 $\begin{align}
  & \left[ p\wedge \left( \sim q\vee r \right) \right]\wedge \left( \sim q\wedge r \right) \\
 & \Rightarrow p\wedge \left( \sim q\wedge r \right) \\
\end{align}$
Then, by using the associative property which states that $\left( a\times b \right)\times c=a\times \left( b\times c \right)$.
Now, by using the same property for logical operators as $a\wedge \left( \sim b\wedge c \right)=\left( a\wedge c \right)\wedge \sim b$ and then by using it, we get:
$p\wedge \left( \sim q\wedge r \right)=\left( p\wedge r \right)\wedge \sim q$
So, we get the final equivalent logical statement of the stamen $\left[ \sim \left( \sim p\vee q \right)\vee \left( p\wedge r \right)\wedge \left( \sim q\wedge r \right) \right]$ is $\left( p\wedge r \right)\wedge \sim q$.

So, the correct answer is “Option A”.

Note: Now, to solve these types of questions we need to know some of the basic conversions of the logical expressions from one form to another. So, one of the form used above which we must be aware of is as:
$\left[ a\wedge \left( \sim b\vee c \right) \right]\wedge \left( \sim b\wedge c \right)=a\wedge \left( \sim b\wedge c \right)$
Moreover, we can use another approach in which we can assume the values of p, q and r and calculate the result of the given question and then calculate the results of all options to get the correct option from them by matching it with the result of the question.