Question

The negation of the inverse of $\sim p \rightarrow q$ is

• A. $\sim p \wedge q$
• B. $\sim q \rightarrow p$
• C. $p \wedge (\sim q)$
• D. $p \wedge q$

Hint:

Save time by writing down the conditional definition directly: $A \rightarrow B$ is equivalent to $\sim A \vee B$. The inverse of $\sim p \rightarrow q$ is $p \rightarrow \sim q$, which means $\sim p \vee \sim q$. Taking the negation of $(\sim p \vee \sim q)$ using De Morgan's Law flips everything directly to $p \wedge q$!

Answer 1

The Correct Option is D

Step 1: Identify the starting statement.
We begin with the conditional $\sim p \rightarrow q$.
Step 2: Recall how to form an inverse.
The inverse of $A \rightarrow B$ negates both parts: $\sim A \rightarrow \sim B$.
Step 3: Build the inverse of our statement.
Negating both parts of $\sim p \rightarrow q$ gives $\sim(\sim p) \rightarrow \sim q$, that is $p \rightarrow \sim q$ after removing the double negation.
Step 4: Recall how to negate a conditional.
A handy rule is $\sim(A \rightarrow B) \equiv A \wedge \sim B$, because an implication only fails when the hypothesis is true and the conclusion is false.
Step 5: Negate the inverse.
$\sim(p \rightarrow \sim q) \equiv p \wedge \sim(\sim q)$.
Step 6: Simplify the double negation.
$\sim(\sim q) = q$, so the expression becomes $p \wedge q$.
\[ \boxed{p \wedge q} \]