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.