Question

The inverse of the proposition $\left( {p \wedge ~q} \right) \Rightarrow r$ is
A. $~r \Rightarrow \left( {~p \vee q} \right)$
B. $\left( {~p \vee q} \right) \Rightarrow ~r$
C. $r \Rightarrow p\left( { \wedge ~q} \right)$
D. None of these

Answer 1

Hint : In propositional calculation, an inverse of a logical implication is an implication in which the premise and conclusion are inverted and negated. If a proposition is in the form of $x \Rightarrow y$ (where x is the premise and y is the conclusion), then its inverse proposition should be in the form of $~y \Rightarrow ~x$ . Use this to find the inverse of $\left( {p \wedge ~q} \right) \Rightarrow r$ .

Complete step by step solution:
We are given to find the inverse of a proposition $\left( {p \wedge ~q} \right) \Rightarrow r$ .
The inverse of a conditional statement can be obtained by swapping and negating the hypothesis and conclusion of the original conditional statement.
Here the given proposition is $\left( {p \wedge~q} \right) \Rightarrow r$ where the hypothesis is $\left( {p \wedge~q} \right)$ and the conclusion is r.
Comparing the given proposition with $x \Rightarrow y$ , we get x as $\left( {p \wedge ~q} \right)$ and y as r.
Inverse of $x \Rightarrow y$ is $~y \Rightarrow~x$
On substituting the expressions of x and y, we get the inverse of $\left( {p \wedge~q} \right) \Rightarrow r$ as $~r \Rightarrow ~\left( {p \wedge ~q} \right)$
$\wedge$ represents ‘and’, ~ represents ‘negation’ and $\vee$ represents ‘or’.
Negation of $\left( {p \wedge ~q} \right)$ is $~p \vee ~\left( {~q} \right) = ~p \vee q$
Therefore, $~r \Rightarrow ~\left( {p \wedge ~q} \right)$ is equal to $~r \Rightarrow ~p \vee q$
Therefore, the inverse of the proposition $\left( {p \wedge ~q} \right) \Rightarrow r$ is $~r \Rightarrow \left( {~p \vee q} \right)$
So, the correct answer is “Option A”.

Note : A proposition is a statement which is either true (denoted as T or 1) or false (denoted as F or 0). True and false are called truth values. A proposition of the form “if p then q” or “p implies q” is called a conditional proposition.
Negation of a statement containing ‘or’ becomes a statement containing ‘and’ and negation of a statement containing ‘and’ becomes a statement containing ‘or’. Negation (logical complement) of a negation results in the original state of the object or statement. Negation is the opposite; negation of negation is the opposite of opposite.