Question

The inverse of the statement $ \left( {p \wedge \sim q} \right) \to p $ is:
A. $ \left( { \sim p \vee q} \right) \to \sim p $
B. $ p \to \left( {p \wedge \sim q} \right) $
C. $ \sim p \to \left( { \sim p \wedge q} \right) $
D. $ \sim p \to \left( {p \wedge q} \right) $

Answer 1

Hint: If a statement is given as $ p \to q $ then the inverse of this statement is given as $ \sim p \to \sim q $ . Use this concept to find the inverse of the statement given in the question.

Complete step-by-step answer:
The inverse of a conditional statement is when both the hypothesis and conclusion are negated; if “IF” part is negated and the “then” part is negated. In geometry the conditional statement is referred to as $ p \to q $ . The inverse is referred to as $ \sim p \to \sim q $ where $ \sim $ stands for NOT or negating the statement.
The statement is given as $ \left( {p \wedge \sim q} \right) \to p $ in the question.
Now take the inverse of the statement given in question according to the definition and rules of taking the inverse of statement.
Now simplify the inverse of the given statement:
 $
\Rightarrow \sim \left( {p \wedge \sim q} \right) \to \sim p \\
\Rightarrow \sim p \vee \sim \left( { \sim q} \right) \to \sim p \\
\Rightarrow \left( { \sim p \vee q} \right) \to \sim p
  $
So, the inverse of the statement $ \left( {p \wedge \sim q} \right) \to p $ is the statement $ \left( { \sim p \vee q} \right) \to \sim p $ .
So, the correct answer is “Option A”.

Note: In this type of question students often make mistakes while writing the inverse and end up getting the wrong answer. Students should remember that $\sim p $ is always of $ p $ and $ p \to q $ means if $ p $ is true the $ q $ have to be true.