Question

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

Answer 1

We will use the property for two elements p and q, that the inverse of $p \to q$ is given by; $\sim p \to \sim q$. Using this on the given proposition, we will get an in the form of $\sim p \to \sim q$. If it further needs to be simplified, we will simplify it or else will leave the answer at this step.

Complete step-by-step answer:
Let us begin with knowing what an inverse actually is. In a logic, an inverse is a type of a conditional sentence, which represents an immediate interference. This interference is made from the other conditional sentence.

It is important to know that every conditional statement has an inverse.

The inverse of $p \to q$ is given by; $\sim p \to \sim q$. Using this property input given proposition we get;
$\left( {p \wedge \sim q} \right) \to r \equiv \sim \left( {p \wedge \sim q} \right) \to \sim r$

Now, we will use the property that, $\sim \left( {p \wedge q} \right) = \sim p \vee \sim q$ .
$\sim \left( {p \wedge \sim q} \right) \to \sim r \equiv \left( { \sim p \vee \sim \left( { \sim q} \right)} \right) \to \sim r$

Now, we will use the property that, $\sim \left( { \sim p} \right) = p$ .
$\left( { \sim p \vee \sim \left( { \sim q} \right)} \right) \to \sim r \equiv \left( { \sim p \vee q} \right) \to \sim r$

Hence, the inverse of the given proposition, $\left( {p \wedge \sim q} \right) \to r$ is $\left( { \sim p \vee q} \right) \to \sim r$. Hence, option (B) is the correct option.

Note: You need to know the meaning of each and every symbol used in the Logic, to be able to solve these kinds of questions. One symbol misinterpreted will lead to a completely wrong answer. This question can also be solved by making a truth tables and then applying the logic. But that method is a lot more time taking, hence if you are thorough with the property used in logics, you should be able to solve these questions in seconds, without any requirement of making lengthy truth tables, in which there is high chances of errors.